• A dot B = a_1 b_1 + a_2 b_2 + a_3 b_3
• A dot B = || A || * || B || cos(theta).

cos(theta) = A dot B / || A || || B || so that

theta = arcCos( A dot B / || A || || B || ).

Two vectors are perpendicular to one another if the angle between them is 90 degrees.  The cosine of 90 degrees is zero, and if the cosine of an angle between 0 and 180 degrees is zero the angle is 90 degrees.  So two vectors are perpendicular if, and only if, their dot product is zero. 

If the dot product of two vectors is zero we say that the vectors are orthogonal.  In two or three dimensions, that means that the angle between the two vectors is 90 degrees.

Section 9.3

1) 9.3.4 Find v dot u when v =<1,-5,0> u =<0,-4,2>.

2) 9.3.6 Find  v dot w when v = 4i + j and w =3i + 2k.

3) 9.3.10 Determine whether v = 5i - 5j + 5k and w = 8i - 10j -2k are orthogonal.

4) 9.3.12 Let v = 4i - 2j + k and w = -2i + j - k. Evaluate (v dot w) * w.

5) 9.3.16 Find the angle between v = 2i +3 k and w = -j + 4k.

6) 9.3.20 Find the scalar and vector projections of v = i - 2j onto w = j - 2k.

7) 9.3.24 Find two distinct unit vectors orthogonal to both v = i + 2j -2k and w = i + j - 2k.

8) 9.3.30 Find x so that v = 2i - xj + 3k and w = -2i + j + xk are orthogonal.

9) 9.3.32 Give the direction cosines and direction angles of v = i - 4j.

10) 9.3.36(b,c,d) Let v = i - j + 4k and w = -i + 3j + 2k. Find cos(theta). Find s such that v is orthogonal to sv - w. Also find t such that v - tw is orthogonal to w

11) 9.3.40 Find the work performed when a force F = (6/11)i - (2/11)j + (6/11)k is applied to an object moving along the line from P(3,5,-4) to Q(-4,-9,-11).

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Section 9.4

1) 9.4.6 Find v X w when v = 10i - 2j + 4k and w =-i -(1/2)j - 3k. (Where X denotes the cross product)

2) 9.4.10 Find v X w when v = sin(theta)i + cos(theta)j and w = -cos(theta)i + sin(theta)j  (theta is any angle).

3) 9.4.12 Find sin(theta) where theta is the angle between v = -i + j and w = -i + j + 2k.

4) 9.4.16 Find a unit vector which is orthogonal to both v = -i + 3j and w = i - j - k

5) 9.4.18 Find a unit vector which is orthogonal to both v = 2i - j and w = 2j - k.

6) 9.4.22 Find the area of the parallelogram determined by the vectors v = 4i + k and w = 4j - k.

7) 9.4.24 Find the area of the triangle with vertices P(2,0,0), Q(1,1,-1), R(3,1,2).

8) 9.4.28 Determine if each of the following products is a vector, scalar, or not defined at all. Explain why.  u X (v X w) , u dot (v dot w), (u X v) dot (w X r). 

9) 9.4.30 Find the area of the parallelepiped determined by u = i - j, v = i - 2k, and w = 4k

10) 9.4.38 Find a number t such that the vectors -i - j, i - (1/2) j + (1/2)k and -2i -2j - 2tk all lie in the same plane.

11) 9.4.39 u = 2i + 2j, v= i-(1/2)j+ (1/2)k, w = i. Compute (u X v) X w and u X (v X w). What does this say about the associativity of the cross product?

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Section 9.5

1) 9.5.4 Find an explicit relationship between x and y by eliminating the parameter in the following equations: x = t, y = 2 - 3t. Sketch the corresponding curve for 0 <= t <= 1.

2) 9.5.8 Find an explicit relationship between x and y by eliminating the parameter in the following equations: x = t^6, y = t^4 -3. Sketch the corresponding curve for -1 <= t <= sqrt(2).

3) 9.5.16 Find an explicit relationship between x and y by eliminating the parameter in the following equations: x = e^-t, y = e^t. Sketch the corresponding curve for -inf <= t <= inf. (inf stands for infinity).

4) 9.5.20 Find the parametric and symmetric equations for the line passing through the points (1,3,-2) and (-1,2,-1).

5) 9.5.22 Find the parametric and symmetric equations for the line passing through the point (-1,-1,0) and parallel to the line (x-3)/4 = (y-1)/3 = (z+3)/2

6) 9.5.26 Find the parametric form of the equation of the line passing through (-3,1,1) parallel to both the xz- and yz-planes.

7) 9.5.28 Find the intersection of the line (x-1)/3 = (y+5)/2 = (z+6)/3 with each of the coordinate planes.

8) 9.5.30 Find the intersection of the line represented by the parametric equations x = 3t + 4, y = 1 - 3t, z = 2t - 7 with each of the coordinate planes (or if it doesn't intersect one, specify which one).

9) 9.5.32 Tell whether the line represented by the parametric equations x = 2-t, y = 3t , z = 3 - 2t and the line represented by x = 5-t, y = -1-3t, z = -3 +4t intersect, are parallel, or if they are skew. If they intersect, give the point of intersection.

10) 9.5.36 Tell whether the line represented by the equation (x-1)/2 = (y-1)/-1 = (z-2)/1 and the line represented by (x+2)/3 = (y+3)/-1 = (z-4)/1 intersect, are parallel, or if they are skew. If they intersect, give the point of intersection.

11) 9.5.40 Find a parametric equation of a circle of radius 4, centered at the origin, and oriented clockwise.

12) 9.5.48 Determine whether the vector v = -(7/3)i - (4/3)j - k is orthogonal to the line passing through the points P(-2,2,7) and Q(1/2,-1/2,9/2).

13) **9.5.52 What can be said about the lines (x-x0)/a1 = (y-y0)/b1 = (z-z0)/c1 and (x-x0)/a2 = (y-y0)/b2 = (z-z0)/c2 if  a1a2 + b1b2 + c1c2 = 0? 

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Section 9.6

1) 9.6.6 Write the equation of the plane 3(x-2) - 2(y-1) - 3(z-5) = 0 in standard form.

2) 9.6.10 Find the equation of the plane containing the point P(-1,3,2) and having normal vector N = 3j - 1k.

3) 9.6.12 Find the equation of the plane containing the point P(0,0,0) and having normal vector N = i.

4) 9.6.14 Find two unit vectors perpendicular to the plane x + 3y - 4z = 2.

5) 9.6.18 Find the distance between the point P(-1,1,-1) and the plane 2x - 3y + z = 4.

6) 9.6.22 Find the distance between the point (-1,2,1) to the plane through the points (0,0,0) , (2,4,8) and (2,1,-1).

7) 9.6.24 Find the distance between the point (-1,2,1) to the plane containing the point (3,3,-2) with normal vector N = -2i + j + 3k.

8) 9.6.30 Find the distance between the point P(1,1,1) and the line (x-2)/3 = (y-1)/2 = (z-3)/4.

9) 9.6.32 Find the distance between the lines (x+1)/-2 = (y+2)-2 = (z+1)/-1 and (x-4)/5 = (y+1)/2 = (z-1)/3

10) 9.6.36 Find the equation of the sphere with center C(-2,7,1) and tangent the the plane x + 4y - 2z = 10.

11) 9.6.50 Find an equation for the plane that contains the point (1,1,-2) and is orthogonal to the line (x-3)/2 = (y+1)/4 = (z-3)/2.

12) 9.6.54 The angle between two planes is defined to be the acute angle betwee the normal vectors of the planes. Find the angle between the planes x - y + 3z = 2 and 2x + y - z = -2. Round the angle to the nearest degree.

13) **9.6.57 Show that the angle between the planes (A1)x + (B1)y + (C1)z + D1 = 0 and (A2)x + (B2)y + (C2)z + D2 = 0 is pi/2 if and only if (A1)A2+ (B1)B2 + (C1)C2 = 0. (In case you don't remember, proving an "if and only if" statement means start with the statement on the left and get to the one on the right and then start with the one on the right and get to the left.)

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Section 9.7

1) 9.7.6 Identify the quadric surface (x^2)/16 + (y^2)/1 -(z^2)/9.

2) 9.7.10 Identify the quadric surface 4y = (z^2)/4 - (x^2)/9.

3) 9.7.16 Identify the quadric surface given by the equation (x^2)/9 - (y^2)/4 - (z^2)/4 = 1. Describe the traces in planes parallel to the coordinate planes (and sketch the graph).

4) 9.7.20 Identify the quadric surface given by the equation 8z^2 = (1/8) + (x^2)/9 + (y^2). Describe the traces in planes parallel to the coordinate planes (and sketch the graph).

5) 9.7.24 For the surface given by the equation y^2 - 4x^2 + 16z^2 = 16. Give the axis of symmetry (either x- or y-axis) and identify the surface (and sketch the surface).

6) 9.7.26 Describe the quadric surface given by the equation 4z = (2x + 4 )^2 + ((2y -1)^2)/9 + 3.

7) 9.7.30 Describe the quadric surface given by the equation ((x-3)^2)/2 - ((y-1)^2)/4 - (z^2-2)/9 = 4.

8) 9.7.32 Describe the curve intersection of the two quadric surfaces 4z = (y^2)/9 - (x^2)/16 and (x^2)/4 + 2(y^2) - 4(z^2)/3 = 1.

9) 9.7.36 What is the intersection of the surfaces z = x^2 + y^2 and y = x?

10) 9.7.38 The line (x = 1 + t , y = 2 - t , z = (1/3)t ) intersects the hyperboloid 4z^2/9 - 4x^2 + y^2/4 =4 at two points. Find the distance between these two points.

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Section 10.1

1) 10.1.4 Find the domain of the vector function F(t) =  (sin t)i +(tan t)j - (sec t)k

2) 10.1.8 Find the domain of F(t) X G(t) when F(t) = t^2 i - (t+2)j + (t-1)k and G(t) = (1/(t+2))i + (t-5)j + sqrt(t) k.

3) 10.1.14 Describe the graph of G(t) = (e^t) i + (t^2) k in R^3

4) 10.1.20 Describe the graph of G(t) = (sin t)i + (cos t)j + (4/3)k

5) 10.1.22 Given F(t)= ti - 5tj +(t^2)k and H(t) = (cos (1-t))i + (e^t)k, find t^2( F(t)) - 3H(t).

6) 10.1.24 Given F(t)= ti - 3tj +(2t^2)k and H(t) = (tan t)i + (e^-t)j, find F(t) X G(t)

7) 10.1.32 Given F(t)= (t)i - 5(e^t)j +(t^3)k, G(t) = ti - (1/t)k and H(t) = (t*sin t)i + (e^-t)j, find H(t) dot [G(t) X F(t)]

8) 10.1.40 Find a vector function F whose graph is the curve given by the following equation x/5 = (y-3)/6 = (z+2)/4.

9) 10.1.44 Find parametric equations for the curve of the intersection (or show that the surfaces do not intersect) for the elliptical cylinder x^2 + 3y^2 =9 and the parabolic cylinder 4z = 2x^2 - 1.

10) 10.1.48 Find the limit as t -> 2 of ((t^4-2)/(t-2))i + ((t^2-4)/(t^2-2t))j + ((t^2 + 3)e^(t-2))k.

11) 10.1.58 Determine all values of t which G(t) = u/(||u||) is defined where u = (t^2)i + sqrt(-t)j.

12) 10.1.60 How many revolutions are made by the circular helix R(t) = (sin t)i + (cos t)j + (3/4)tk in a vertical distance of 12 units.

13) **10.1.62 Define `dH = H(t + `dt) - H(t) where `dt is a small change in the parameter t. Show that `d(F X G)(t) = F(t + `dt) X `dG(t) + `dF(t) X G(t). (Here where I use `d the book uses a capital delta. If you need help look at the hint in the book for this problem, it will give you a very good start.)

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Section 10.2

1) 10.2.2 Find F' for F(s) = ((s+1)i + (s^3)j - (s^2)k) + ((s^2)i + sj + (3s+3)k).

2) 10.2.4 Find F' for F(t) = (cos t)(3i + (tan t)j + 3(sec t)k)

3) 10.2.8 Find both F' and F'' for F(t) = (4sin^2 t)i + (9cos^2 t)j + tk

4) 10.2.10 Differentiate g(x) = <sin x, (1/x) ,-x> dot <csc x, -(x^2), 3x> with respect to x.

5) 10.2.16 Given the position vector of a particle R(t) = (cos t)i + tj + (4 sin t)k, find the particle's velocity and acceleration vectors and then find the speed and direction of the particle at t = pi/2.

6) 10.2.20 Find the tangent vector to the graph of F(t) = (-ti + (t^2)j - (t^3)k)/(1- t) at t = 0 and at t =2.

7) 10.2.24 Find parametric equations for the tangent line to the graph of the vector function F(t) = <cos t, sin t, e^t> at the point corresponding to t = pi.

8) 10.2.26 Find Int(<sin t, cos t, t^2> dt) (Where Int( f(t) dt) is the integral of f with respect to t)

9) 10.2.28 Find Int((e^t)*<t,4t^2,sin t> dt)

10) 10.2.32 Find the position vector R(t) given the velocity vector v(t) = 3(t^2)i + (e^t)j + sqrt(t/2)k and the initial position R(0) = 3i - 2j - k.

11) 10.2.36 Find the velocity and position vectors given the acceleration vector `A(t) = 4(t^2)i - 2 sqrt(t) j + 5(e^3t)k, initial position R(0) = 2i + j -3k and initial velocity v(0) = 4i + j + 2k.

12) 10.2.48 Find a value of a such that the following equality holds. Int( (t(sqrt(1+t^2)) i + (1/(1+t^2))j dt, 0, a) = (2sqrt(2) - 1)i - pi/4 j. (Int(f(t) dt, a, b) in the integral of f(t) with respect to t from a to b).

13) 10.2.50 F(t) = e^(-kt)i + e^(kt)k. Show that F and F'' are parallel.

14) **10.2.60 If G = F dot (F' X F'') then what is G' and what does it mean?

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Section 10.3

1) 10.3.2 Find the time of flight t_f (to the nearest tenth of a second) and the range Rf (to the nearest unit) of a projectile fired (in a vaccum) from ground level at `alpha = 60 degrees and v0 = 75 ft/s. Assume that g = 32 ft/s^2.

2) 10.3.6 Find the time of flight Tf (to the nearest tenth of a second) and the range Rf (to the nearest unit) of a projectile fired (in a vaccum) from ground level at `alpha = 65.54 degrees and v0 = 19.07 m/s. Assume that g = 9.8 m/s^2.

3) 10.3.10 An object is moving along the curve x = -cos t , y = sin t in the plane. Find its velocity and acceleration in terms of the unit polar vectors u_r and u_theta.

4) 10.3.14 An object is moving along the curve r = 1/(1 - sin(theta)), theta = t - pi/2.  Find its velocity and acceleration in terms of the unit polar vectors u_r and u_theta.

5) 10.3.20 A baseball hit at a 35 degree angle from 3.5ft above the ground just goes over the 8ft-fence 400ft from home plate. How fast was the ball travelling, and how long did it take the ball to reach the wall?

6) 10.3.22 If a shotputter throws a shot from a height of 5.5t and an angle of 53 degrees with initial speed 28 ft/s. What is the horizontal distance of the throw?

7) A .25lb paddleball attached to a string is swung in a circular path with a 1.5ft radius. If the string will break under a force of 3lb, find the maximum speed the ball can attain withouth breaking the string

8) A child running along level ground at the top of a 40ft high vertical cliff at a speed of 15ft/s, throws a rock over the cliff into the sea below. If the rock is released 10 ft from the edge and at an angle of 45degrees, how long does it take the rock to hit the water and how far away from the base of the cliff does it hit?

9) A particle moves along the polar path (r, theta) where r(t) = 4 - 2cost and theta(t) = t^2. Find V(t) and A(t) in terms of u_r and u_theta.

10) **A gun is fired with muzzle speed 700ft/s at an angle of 20degrees. It overshoots the target by 60 ft. If the target is moving away from the fun at a constant speed of 15ft/s and the gunner takes 30 seconds to reload, at what angle should the second shot be fired with the same muzzle speed?

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The unit tangent vector is the unit vector in the direction of the velocity.

The tangential component of the acceleration is the projection of the acceleration vector on the unit tangent vector.

The normal component of the acceleration is the acceleration, minus its component in the direction of the unit tangent vector. 

The unit normal vector is the unit vector in the direction of the normal component of the acceleration vector.  A vector parallel to the normal vector can also be obtained by taking the derivative of the unit tangent vector.

The unit binormal vector is the cross product of the unit tangent and unit normal vectors.

The speed of a point whose position function is R(t) is the magnitude of the velocity vector.

Section 10.4

1) Find T(t) and N(t) when R(t) =( t^2 cost) i + (t^2 sint) j.

2) Find T(t) and N(t) when R(t) = (e^-2t cost)i + (e^-2t sint )j + e^-2t.

3) Find the length of R(t) = (7/3)t^3i + 2t^3j + 4t^3k on the interval [0,1].

4) Find the curvature of the plane curve y = sin (-3x) at x = pi/2.

5) Express R(t)= <sin2t, cos2t> in terms of the arc length parameter s measured from the point t = 0 in the direction of increasing t.

6) Let u and v be constant nonzero vectors. Show that the line given by R(t) = u + tv has curvature 0 at each point.

7) Let C be the curve given by R(t) = (1-cos t)i + (t-sin t)j + (4sin(t/2))k

• Find the unit tangent vector T(t) to C
• Find dT/ds and the curvature k(t)

8) Find the maximum curvature for the curve y = e^3x.

9) Find the curvature of R(t) = (3t-3sin t)i + (cost)j + 3k using both cross derivative form and two derivatives form.

10) Find the curvature of r = 1 + sin(theta) for theta in [0,2pi] using the polar form.

11) Let C be a smooth curve in R^2 described by the parametric equations x = x(t) and y = y(t).

• **Derive the parametric form of the curvature using a form which has already been proven in the text. (Hint: How else can you think of parametric equations for a function?)
• Find the curvature of the curve described by x = 1 - sin t, y = 7 + cos t.

12)** If T and N are the unit tangent and normal vectors on the trajectory of a moving body, we can define B = T X N to be the unit binormal vector. A coordinate system with three planes can be made at each point with these vectors since they are mutually orthogonal.

• Show that T is orthogonal to dB/ds (Hint: Differentiate B dot T)
• Show that B is orthogonal to dB/ds (Hint: Differentiate B dot B)
• Show that dB/ds = -(tau)N for some constant tau. (We call the constant tau the torsion of the trajectory)
• **Prove the Frenet-Serret formulas: (dT/ds = kN, dN/ds = -kT + tauB, dB/ds = -tauN). Where k is the curvature and tau = tau(s) is a scalar function.

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Section 10.5

1) Find the tangential and normal components of an object's acceleration which has the position vector R(t) = <2 cos t, 5 sin t>.

2) Find the tangential and normal components of an object's acceleration which has the position vector R(t) = <3/5 cos t, 4/5(1+sin t), cos t>.

3) If V(0) = -2i - 3j and A(0) = 2i + 3j, what is A_T and A_N at t = 0?

4) If V(0) = <5,-2,4> and A(0) = <1,3,-9>, what is A_T and A_N at t = 0?

5) ||V|| = sqrt(e^-t + t^4). What is A_T at t = 0?

6) Where on the trajectory of R(t) = <2t^2 - 5t, 5t + 2, 4t^2> is the speed maximized and minimized?

7) An object moves with a constant angular velocity w around the circle x^2 + y^2 = r^2 in the xy-plane.

• Find a parameterization for the circle.
• Compute the tangential and normal acceleration for the object.

8) A car which weighs 3,000lbs moves along the elliptic path 1600x^2 + 100y^2 = 1, where x and y are measured in miles. If the car travels at the constant speed of 30mi/hr, how much frictional force is required to keep it from skidding as it turns at (1/40,0)? What about at (0,1/10)?

9) ** Consider the vector function R(t) = <3 sin t, 4t, 3 cos t>.

• Evaluate V(t) = R'(t), N(t), and A(t) = R''(t) when t = 1.
• Find the vector projection of A(1) onto V(1). Denote this proj_V(1) (A(1)).
• Find the vector projection of A(1) onto N(1). Denote this proj_N(1) (A(1)).
• What is the sum of proj_V(1) (A(1)) and proj_N(1) (A(1)).
• How does proj_V(1) (A(1)) relate to A_T when t = 1.
• How does proj_N(1) (A(1)) relate to A_N when t = 1.

10) ** Use the method described in the previous problem to find A_T and A_N for R(t) = <cos t, sin t>.

11) ** Let B = T X N when T and N are the unit tangent and normal vectors to a curve C with position vector R. Show that dB/ds = T X (dN/ds).

12) An object connected to a string of length r is spun counterclockwise in a circular path in a horizontal plane. Let w be the constant angular velocity of the object.

• Show that the position vector of the object is R(t) = (r cos wt) i + (r sin wt) j.
• Find the normal component of the object's acceleration.
• If w is doubled, what is the effect on the normal component of acceleration?
• What is the effect on A if w is unchanged but the length of the string is doubled?
• What happens to A is w is doubled but r is halved?

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Section 11.1

1) Let f(x,y,z) = x^2*y*e^3x + (x - y + z)^2. Find the following expressions.

• f(0,0,0)
• f(1,-1,1)
• f(-1,1,-1)
• d/dx(f(x,x,x))
• d/dy(f(1,y,1))
• d/dz(f(1,1,z^2))

2) Find the domain and range of the function f(x,y) = x/ sqrt(x- y^2).

3) Find the domain and range of the function f(u,v) = sqrt(u cos v).

4) Find the domain and range of the function f(x,y) = e^((x+1)/(y-3)).

5) Sketch and describe the level curves f(x,y) = C for C >= 0 for f(x,y) = y/x.

6) Sketch and describe the level surface f(x,y,z) = 1 when f(x,y,z) = 2x^2 + 2z^2 - y.

7) Sketch and describe the traces of the quadric surface z = x^2/25 - y^2/9.

8) Sketch and describe the graph of the function f(x,y) = y - 3.

9) Sketch and describe the graph of the function f(x,y) = -sqrt(1 - x^2 - y^2).

10) According to the ideal gas law, PV = kT where P is pressure, V is volume, T is temperature, and k is some constant. Suppose a tank contains 3500in^3 of some gas at a pressure of 24ln/in^2 when the temperature is 270K.

• Determine k for this gas.
• Express T as a function of P and V using the k found in the previous step and describe the isotherms.

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Section 11.2

1) Find the limit as (x,y) -> (0,0) of (5x^2 - 2xy + 3y^2 -9) or explain why the limit does not exist.

2) Find the limit as (x,y) -> (e,0) of (ln(x^2 + y^2)) or explain why the limit does not exist.

3) Find the limit as (x,y) -> (0,0) of (x^2y^2/(x^2 + y^2)) or explain why the limit does not exist.

4) Find the limit as (x,y) -> (a,a) of ((x^4-y^4)/(x^2 - y^2)) where a is some constant, or explain why the limit does not exist.

5) Find the limit as (x,y) -> (0,0) of (y - x)/sqrt(x^2 + y^2) or explain why the limit does not exist.

6) Find the limit as (x,y) -> (0,0) of [1 - (sin(x^2 + y^2)/(x^2 + y^2)] or explain why the limit does not exist.

7) Explain why the limit as (x,y) -> (0,0) of f(x,y) = (x^2 + y)/(x^2 + y^2) does not exist.

8) Let f(x,y) = (y^2 - x^2)/(x^2 + y^2) for all x and y such that (x,y) isn't (0,0)

• Find the limit as (x,y) -> (2,1) of f(x,y).
• Show that f(x,y) does not have a limit at (0,0).

9) **Let f(x,y) = cos(x^2 + y^2)/(x^2 + y^2) when (x,y) is not (0,0). Using polar coordinates to see for what value of f(0,0) is f(x,y) continuous at (0,0).

10) **Either show that the following statement is true or give a counterexample: If the limit as y -> 0 of f(0,y) is 0, then lim (x,y) -> (0,0) of f(x,y) is also 0.

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Section 11.3

1) Find f_x, f_y, f_xx, and f_yx when f(x,y) = (x + xy + y)^4. (f_x is the first partial with respect to x, f_yx is the partial with respect to y first and then x)

2) Find f_x and f_y when f(x,y) = xy^4*arctan(y).

3) Find f_x and f_y when f(x,y) = arccos(xy)*e^(x+y).

4) Find f_x, f_y, and f_z when f(x,y,z) = xz*e^y.

5) Find f_x, f_y, and f_z when f(x,y,z) = cos(yz + x).

6) Determine z_x and z_y by differentiating the expression 4x^2 + 2y^2 + 3z^2 = 9 implicitly.

7) Let f(x,y) = (x^2 + y^2)/(xy), P = (2, -1, -5/2)

• Find the slope of the tangent line of the graph of f parallel to the xz-plane at the point P.
• Find the slope of the tangent line of the graph of f parallel to the yz-plane at the point P.

8) Let f(x,y) = x*ln(x+y^2), P = (e, 0 ,e)

• Find the slope of the tangent line of the graph of f parallel to the xz-plane at the point P.
• Find the slope of the tangent line of the graph of f parallel to the yz-plane at the point P.

9)** Determine f_x and f_y for f(x,y) = Int( (e^t + 3t) dt, x^2, 2y). (Remember the second fundamental theorem of calculus.)

10) A function is said to be harmonic on the open set S if f_xx + f_yy = 0 for all (x,y) in S. Show that the following functions are harmonic for the given S.

• f(x,y) = 2ln(x^2 + y^2). S = xy-plane with point (0,0) removed.
• f(x,y) = e^y*cos(x). S = xy-plane.

11) For the two following functions, show that f_xy = f_yx.

• f(x,y) = cos(yx^2).
• f(x,y) = (cos^2(x))*(cos y).

12) In physics the wave equation is given by z_tt = c^2 * z_xx and the heat equation is given by z_t = c^2 * z_xx. In the two following cases, see if z satisfies the wave equation, the heat equation, or neither.

• z = sin(2t)*sin(2ct).
• z = (e^(-t))(sin (x/c) + cos(x/c).

13) The Cauchy-Riemann equations are u_x = v_y and u_y = -v_x. Do the equations u = e^x*cos y and v = e^x* sin y satisfy the Cauchy-Riemann equations?

14) **Given the central conic p(x,y) = ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 with ab - h^2 not equal to 0. Show that the center is the intersection of the lines f_x = 0 and f_y = 0.

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Section 11.4

1) Give the standard form equation for the tangent plane to the given surface at the point P_0.  z(x,y) = 5 - x^2 -y^2 at the point P_0 = (1,1,3).

2) Give the standard form equation for the tangent plane to the given surface at the point P_0.  z(x,y) = ln(x^2 + y^2) at the point P_0 = (e,0,2).

3) Find the total differential of f(x,y) = 7xy + 3x^2*y + 18x + 9y + 2.

4) Find the total differential of f(x,y,z) = 2xzy^3*cos(xy)*sin(z).

5)** Show that the function f(x,y) = e^(2x + y^2) is differentiable for all (x,y) in R^2.

6) Use an incremental approximation to estimate the function f(x,y) = 2x^3 + 4y^2 at f(1,01, 2.03).

7) Use an incremental approximation to estimate the function f(x,y) = cos(xy) for f(sqrt(pi) + .01, sqrt(pi) - .01).

8) Find the equation of all horizontal tangent planes to the surface z = 4 - x^2 - y^2 + 6x.

9)** Show that if x and y are sufficently close to zero and f is differentiable at (0,0), then f(x,y) ~= f(0,0) + x*f_x(0,0) + y*f_y(0,0).

• Use the formula in the previous part to show that 1/(1+x-y) ~= 1-x+y for small x and y.
• If x and y are sufficently close to zero, what is the approximate value of the expression 1/((x+1)^2 + (y+1)^2).

10) It is know that the period T of a simple pendulum with small oscillations is modeled by T = 2*pi*sqrt(L/g) where L is the length of the pendulum and g is the acceleration due to gravity. For a certain pendulum it is known that L = 3.01ft and g = 32.2ft/s^2. What is the approximate error in calculating T by using L = 3ft and g = 32ft/s^2?

11) A football has the shape of an ellipsoid x^2/9 + y^2/36 + z^2/9 = 1 with dimensions in inches. If you know that the leather shell is 1/8in thick, use differentials to estimate the volume of the leather shell. The volume of an ellipsoid with formula x^2/(a^2) + y^2/(b^2) + z^2/(c^2) = 1 is 4/3*pi*abc.

12) Compute the total differentials of x/(x-y) and y/(y-x). Why are these two differentials equal?

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Section 11.5

1) Let z = f(x,y) = 2xy + x^2 where x = 3t^2 and y = -4t.

• Find dz/dt after finding z explicitly in terms of t.
• Use the chain rule for one parameter to find dz/dt.

2) Let z = f(x,y) = xy + 1 where x = cos 3t and y = cot 3t.

• Find dz/dt after finding z explicitly in terms of t.
• Use the chain rule for one parameter to find dz/dt.

3) Let F(x,y) = x^2 + y^2 where x(u,v) = u cos(v) and y(u,v) = u + v^2. Let z = F(x(u,v),y(u,v)). Find z_u and z_v in the following ways.

• Expressing z explicitly in terms of u and v.
• Apply the chain rule for two independent parameters.

4) Let F(x,y) = ln(xy) where x(u,v) = e^(uv) and y(u,v) = e^(uv + 1). Let z = F(x(u,v),y(u,v)). Find z_u and z_v in the following ways.

• Expressing z explicitly in terms of u and v.
• Apply the chain rule for two independent parameters.

5) Write out the chain rule for the function w = f(x,y) where x = x(s,t,u,v) and y = y(s,t,u,v).

6) Find dw/dt, where w = xe^(yz^2) and x = cos t, y = sin t, z = tan t.

7) Find w_r where w = e^(x - y + 3z^2) and x = r + t - s, y = 3r - 2t, z = sin(rst).

8) Find dy/dx when  x cos y + y sin x = x.

9) Let x sin z = x + 2y. Find z_xy, z_xx, and z_yy.

10) Let f(x,y) be a differentiable function of x and y and let x = r cos(theta) and y = r sin(theta) for r > 0 and 0 < theta < 2pi.

• If z = f(x(r,theta) , y(r,theta)), find z_r and z_theta.
• **Show that (z_r)^2 + 1/r^2(z_theta)^2 = z_x^2 + z_y^2.

11) The dimensions of a rectangular box are linear funcitons with respect to time: l(t), w(t), and h(t). The length and width are increasing at 3 in/sec and the height is decreasing at 2 in/sec.

• Find the rate at which the volume V and the surface area S are changing with respect to time.
• If l(0) = 10 ,w(0) = 8, and h(0) = 20, is V increasing or decreasing when t = 5. What about S?

12) The combined resistance of three resistors R is given by the formula 1/R = 1/(R1) + 1/(R2) + 1/(R3). Suppose that at a certain instant R1 = 150 ohm, R2 = 300 ohm, and R3 = 450 ohm. R1 and R3 are decreasing at a rate of 3 ohm/sec and R2 is increasing at a rate of 4 ohm/sec. How fast is R changing at this instant and is it increasing or decreasing?

13) Find z''(theta) where z is a twice differentiable function of theta and z can be written as z = f(cos(theta), sin(theta)). (Hint: What substitution do you make to go from rectangular coordinates to polar coordinates?)

14)** The Cauchy-Riemann equations are u_x = v_y and u_y = -v_x. Show that if x and y are expressed in terms of polar coordinates, the Cauchy-Riemann equations become u_r = 1/r*v_theta and v_r = -1/r*u_theta.

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Section 11.6

1) Find grad(f) when f(x,y) = ln(x^2 + y). (grad(f) is the gradient of f).

2) Find grad(f) when f(x,y,z) = e^(x+y+z).

3) Find grad(f) when f(x,y,z) = (xz-1)/(y -x).

4) Find the directional derivative of f(x,y) = x^2 + xy at the point (1, -1) in the direction of the vector v = i - j.

5) Find a unit vector which is normal to the surface given by the equation 2 = x^3 + 2xy^2 + 3y - z at the point P = (1,1,1). Also find the equation of the tangent plane at this point using this information.

6) Find a unit vector which is normal to the surface x^2 + y^2 + z^2 = 1 at the point P = (a,b,c). Find the equation of the tangent plane at this point.

7) Find the direction from the point P = (1,e,-1) which the function f(x,y,z) = z ln (y/x) increases the most rapidly and compute the magnitude of the greatest rate of increase.

8) Find the direction from the point P = (a,b) which the function f(x,y) = ax + by + c increases the most rapidly and compute the magnitude of the greatest rate of increase.

9) Find a unit vector which is normal to the point P = (x0,y0) for the hyperbola x^2/a^2 - y^2/b^2 = 1.

10) Find the directional derivative of f(x,y) = x^2 - 2xy + y^2 at the point P = (1,-1) in the direction toward the origin.

11) Let f have continuous partial derivatives, and assume the maximal directional derivative of f at (0,0) is equal to 100 and attained in the direction of (-4/5, 3/5). Find grad(f) at (0,0).

12)** Let T(x,y) = 1 - x^2 - 2y^2 be the temperature at each point in a metal sheet. A heat-loving bug is placed in the plane at the point P = (-1,1). Find the path that the bug should take to be as warm as possible. Assume at each point on the path, the tangent line will point in the direction at which T increases most rapidly.

13) A particle P1 with mass m1 is located at the origin, and a particle P2 with mass 1 unit is located at the point (x,y,z). According to Newton's law of universal gravitation, the force P1 exerts on P2 is modeled by F = -G(m1(xi + yj + zk))/r^3 where r is the distance between P1 and P2 and G is the gravatational constant.

• Starting from the fact that r^2 = x^2 + y^2 + z^2, show that d/dx*(1/r) = -x/r^3, d/dy*(1/r) = -y/r^3 and d/dz(1/r) = -z/r^3. (Here d/dx denotes partial with respect to x)
• The function V = -G*m1/r is called the potential energy function for the system. Show tha F = -grad(V).

14)** Suppose that u and v are unit vectors and that f has continuous partial derivatives. Show that D_(u+v)(f) = 1/||u+v|| * D_u(f) + D_v(f). (D_u(f) is the directional derivative of f in the direction of u).

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Section 11.7

1) Consider f(x,y) = 3x^2 - 5xy + y^2 + 3. Find the critical points, and classify each point as either a relative maximum, relative minimum, or saddle point.

2) Consider f(x,y) = e^(-y) cos(x). Find the critical points, and classify each point as either a relative maximum, relative minimum, or saddle point.

3) Consider f(x,y) = e^(-(x^2 + y^2)). Find the critical points, and classify each point as either a relative maximum, relative minimum, or saddle point.

4) Consider f(x,y) = x^2 - y^2 + xy/16. Find the critical points, and classify each point as either a relative maximum, relative minimum, or saddle point.

5) Consider f(x,y) = x^2 +  2xy + y^2. Find the absolute extrema on the closed bounded set S = {(x,y) | x^2 + y^2 <= 1}.

6) Consider f(x,y) = x^3 - 2xy + y^2 + 3x. Find the absolute extreme on the closed bounded set S = {(x,y) | 0 <= x,y <= 2}.

7) Find the least squares regression line for the set of points {(4,-2), (3,-1), (0,0), (-1,3), (-2,1), (-3,2)}.

8) Find three positive numbers whose sum is 38 and whose product is as large as possible.

9) A particle of mass m in a rectangular box with dimensions x,y,z has ground state energy E(x,y,z) = k^2/(8m) * (1/x^2 + 1/y^2 + 1/z^2) where k is a physical constant. Find the values of x,y,z that minimize the ground state energy in the following cases.

• If the volume of the box is 30 and x = y.
• If the volume of the box is 45.
• If the volume of the box is V.

10)** We can use the exponential and logarithim to help us to linearlize data that does not tend to change linearly. The following problems will demonstrate this.

• Suppose we have y = kx^m. Show that by taking the natural logarithm of this equation we obtain a linear relationship Y = K + mX. Explain these new variables and the constant K.
• The data that follows relates the periods of revolution, t (in days), of the six inner planets and to their semimajor axis, a (in 10^6 km). Kepler conjectured that the relationship is t = ka^m for some k and m. Transform the data as in the first part and find k and m.

t-data : (87.97, 224.7, 365.26, 686.98, 4332.59, 10759.2)  a-data : (58, 108, 149, 228, 778, 1426).

11) Consider these following functions which D = 0 at a critical point. Show that the function has a saddle point, relative minimum, or relative maximum as specified.

• Show f(x,y) = x^4 - y^4 has a saddle point at (0,0).
• Show g(x,y) = x^2*y^2 as a relative minimum at (0,0).
• Show h(x,y) = x^3 + y^3 as a relative maximum at (0,0).

12)** Consider the function f(x,y) = (x - y^2)(x - 2y^2). Discuss the behavior of this function at (0,0).

13)** Prove the second partials test. As a start compute the second directional derivative of f in the direction of the unit vector u = hi + kj and complete the square.

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Section 11.8

1) Maximize the function f(x,y) = x^2 + y^2 subject to the constraint x + y = 12.

2) Maximize the function f(x,y) = e^(xy) subject to the constraint x^2 + y^2 = 1.

3) Maximize the function f(x,y,z) = xyz subject to the constraint x + 2y + 3z = 12.

4) Maximize the function f(x,y,z) = x^2 + y^2 + z^2 subject to the constaint x^2 + 2y^2 + 4z^2 = 4.

5) Find the minimum and maximum values of f(x,y,z) = x + 2y - z on the sphere x^2 + y^2 + z^2 = 100.

6) Find the minimum and maximum distance from the origin to the ellipse 5x^2 -6xy + 5y^2 = 4.

7) Post office regulations specify that a box can be mailed parcel post only if the sum of its length and girth (if dimensions are x,y,z, girth is 2x + 2y and z is length) does not exceed 108 in. Find the maximum volume of such a package.

8)** Heron's formula says that the area of a triangle with sides length a,b,c is given by sqrt(s(s-a)(s-b)(s-c)) where s = 1/2*(a+b+c). Show that given a fixed parameter P, the equilateral triangle has the largest area.

9) Find the maximum of f(x,y,z) = xyz subject to the constraints x^2 + y^2 = 3 and y = 2x.

10)** A farmer wants to build a metal silo in the shape of a right circular cylinder with a right circular cone on top. The bottom of the silo will be a concrete slab. What is the least amount of metal that can be use if the silo is to have a fixed volume V0.

11) Use Lagrange multipliers to optimize f(x,y) = 2x + 2y subject to the constrain xy = 1/4. In this case you will recieve two candidates for an extremum. Explain.

12) Show that the cost function C(x,y) = px + qy is minimized subject to the fixed construction level Ax^(alpha)*y^(beta) = k with alpha + beta = 1 when x = k/A(alpha*q/(beta*p)^(beta), y = k/A(beta*p/(alpha*q))^(alpha).

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Section 12.1

1) Evaluate the iterated integral Int[ Int( x^2 + 2xy + y^2 dy, 0,2) dx, 0,1].

2) Evaluate the iterated integral Int[ Int( 1/(x+y)^3 dy, -1, 2) dx, 2,3].

3) Evaluate the double integral over R with respect to A of 7  where R: 2 <= x <= 4, 1 <= y <= 5.

4) Evaluate the double integral over R with respect to A of x/4  where R: 1 <= x <= 8, -2 <= y <= 0.

5) Compute the double integral over R with respect to A of (1 + x^2)/(4 + y^2)  where R: 0<= x <= 1, 0 <= y <= 1.

6) Compute the double integral over R with respect to A of cos(x+y)  where R: 0 <= x <= pi/2, 0 <= y <= pi/2.

7) Find the volume of the solid bounded above by the graph of z = f(x,y) and below by the xy-plane when f(x,y) = ye^(xy) and R: 0 <= x <= ln2, 0 <= y <= 1.

8) Let R be a rectangular region within the boundary of a certain city defined by R: -2 <= x <= 3, -1 <= y <= 1. The units are in miles and (0,0) is the city center. Assume the population density is 13*e^(-0.08)r thousand people per square mile and r = sqrt(x^2 + y^2). Give the double integral which will model the total population of this region, do not solve the integral however.

9) Compute the double integral over R with respect to A of ln(sqrt(x))/xy  when R is the rectangle 1 <= x <= e, 1 <= y <= 4.

10) Use a grid with 16 cells to approximate the area under the surface of 4 - x^2 - y^2 an the rectangle R: 0 <= x,y <= 1.

11) **Let f be a function with continuous second partial derivatives over a rectangular region R with vertices (x1,y1), (x1,y2), (x2,y1) and (x2,y2) where x1 < x2 and y1 < y2. Show using the Fundamental Theorem of Calculus that the double integral over R with respect to A of f_xy is equal to f(x1,y1) - f(x2,y1) + f(x2,y2) - f(x1,y2).

12) **Let f(x,y) = (y-x)/(x + y)^3 and define R: 0 <= x,y <= 1. Find the double integral of f(x,y) over R with respect to A using both orders of differentiation. Why is it that the two integrals have different answers and why do they not violate Fubini's theorem.

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Section 12.2

1) Evaluate the double integral Int[ Int( dy, x^2, 4x) dx, 0, 4].

2) Evaluate the double integral Int[ Int((2x + 3y) dy, -1, x) dx, -1, 1].

3) Evaluate the double integral Int[ Int((x^2* e^xy) dy, 0, x) dx, 0 , 1].

4) Evaluate the double integral Int[ Int(sqrt(1 + sin^2(x)) * sin x dx, arccos y, pi/2) dy, 0, 1].

 5) Integrate 4x over D with respect to A using a double integral where D is the region bounded by 4 - x^2, y = 3x, and x = 0.

6) Integrate 1/(y^2 + 1) over D with respect to A using a double integral where D is the triangle bounded by x = 2y, y = -x, and y = 2.

7) Compute the following integral with the given order of integration and with the order changed: Int[ Int(4x^3 dy, 0, sqrt(x)) dx, 0, 4].

8) Give an equivalent integral with the order of integration reversed. Int[ Int( f(x,y) dy, arctan x, pi/4) dx, 0, 1]

9)** Set up a double integral for the solid remaining when a square hole with side 3 is drilled through a sphere with radius 2.

10) Give two different ways to set up the integral of the area of the region D where D is the region in the first quadrant of the xy-plane bounded by y = 4/x^2 and y = 5 - x^2. Evaluate one of these integrals.

11) Reverse the order of integration in Int[ Int( f(x,y) dy, x, x^3) dx, 1,2] + Int[ Int(f(x,y) dy, x, 8) dx, 1,2].

12)** Let f(x,y) be continuous on a region D and m <= f(x,y) <= M for all (x,y) in D.

• Show that mA <= Double integral over D of f(x,y) with respect to A <= MA, where A is the area of D.
• Use the previous step to estimate the value of the double integral of e^(x cos y) over D with respect to A where D is the triangle with vertices (0,-1),(0,2),(1,0).

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Section 12.3

1) Evaluate the double integral Int[ Int(sqrt(1-r^2)*r dr, 1,2) dtheta, 0, pi/2]. Sketch and describe the region over which this integral is taken.

2) Evaluate the double integral Int[ Int(dr, 0, 1 + sin(theta)) dθ, 0 , pi]. Sketch and describe the region over which this integral is taken.

3) Use a double integral to find the area of the region described by 0 <= r <= 3 cos(5theta).

4) Use a double integral to find the area of the region described by 1 <= r <= 3 sin(θ).

5) Use a double integral to find the area bounded by the parabola r = 5/(1 + cosθ) and the lines θ = 0, θ = π/3 and r = 5/2* secθ.

6) Use polar coordinates to find the double integral of f(x,y) = x^2 + y^2 over D with respect to A where D is the circular disk defined by x^2 + y^2 <= a^2 for some a > 0.

7) Use polar coordinates to find the double integral of f(x,y) = ln( a^2 + x^2 + y^2) over D with respect to A where D is the circular disk defined by x^2 + y^2 <= a^2 for some a > 0.

8) Use polar coordinates to find the double integral of f(x,y) = sqrt(x^2 + y^2) over D with respect to A where D is the region inside the circle x^2 + (y-1)^2 = 1 in the fourth quadrant.

9) Use polar coordinates to find the double integral of f(x,y) = 1/ sqrt(9 - x^2 - y^2) over D with respect to A where D is the region inside the circle x^2 + y^2 = 4 in the first quadrant.

10) Find the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 4 and below by the parabola 3z + x^2 + y^2.

11) Use polar coordinates to find to integrate x^2 + y^2 over the region D which the upper part (i.e., the part above the x axis) of a "washer"-shaped region (annulus) with inner radius 1 and outer radius 4.

12) **Let I = Int(e^(-x^2) dx, -inf, inf). Show that I = sqrt(pi). Note that Int(e^(-x^2) dx, -inf, inf) * Int(e^(-y^2) dy, -inf, inf) = Int[ Int(e^-(x^2 + y^2) dx, -inf, inf) dy, -inf, inf].

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Section 12.4

1) Find the surface area of the portion of the plane 4x + y + x = 9 that lies in the first octant.

2) Find the surface area of the portion of the surface z = y^2 that lies over the triangular region in the plane with vertices (0,0,0), (1,0,0) and (0,1,0).

3) Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 36 that lies above the plane z = 4.

4) Find the surface area of the portion of the parabola z = 9 - x^2 - y^2 that lies above the xy-plane.

5) Find the surface area of the portion of the cylinder x^2 + y^2 = 16 that lies within the cylinder x^2 + z^2 = 16 .

6) Find a formula for the surface area of the frustrum of the cone z = 9sqrt(x^2 + y^2) between the planes z = h1 and z = h2 where h1 > h2.

7) Find the surface area of the portion of the cylinder x^2 + z^2 = 9 which lies above the triangle with vertices (0,0,0), (1,1,0), (1,0,0).

8) Set up (but do not solve) the double integral for the surface area of the surface given by e^(-x)*cos y over the disk  x^2 + y^2 <= r^2

9) Compute the magnitude of the fundamental cross product for the surface parametrically described by R(u,v) =(2u sin v)i + (2u cos v)j + (u^2 sin2v)k.

10) Consider the surface S which is a sphere of radius a.

• Find a parameterization for S.
• Using this parameterization find the fundamental cross product and its magnitude.
• Find the surface area for S.  Does it match up with the familiar formula for surface area of a sphere?

11) Find the surface area of the torus which is parametrically described as R(u,v) = (a + b cos v)cos u i + (a + b cos v)sin u j + vk. Where 0 < b < a, 0 <= u,v <= 2pi.

12)** Let S be the surface defined by f(x,y,z) = C, and let R be the projection of S onto a plane. Show that the surface area of S can be computed by the double integral of (||grad(f)||/(|grad(f) dot u|) with respect to A where u is a unit veter normal to the plane containing R and grad(f) dot u is non-zero.

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Section 12.5

1) Compute the iterated triple integral Int[ Int[ Int(dx,4,5) dy,-2,1]dz,1,3].

2) Compute the iterated triple integral Int[ Int[ Int(x^2*y sin(xyz) dz, 0,1) dy,0,1] dz,0, pi].

3) Evaluate the triple integral of xz + 2yx over D with respect to V where D is the box 2<= x <= 4, 1 <= y <= 3, -1 <= z <= 1.

4) Evaluate the triple integral of  x^2*y over D with respect to V where D is the tetrahedrom with vertices (0,0,0), (3,0,0), (0,2,0), (0,0,1).

5) Evaluate the triple integral of xy over D with respect to V where D is the solid in the first octant bound by the hemisphere sqrt(4-x^2 - y^2) and the coordinate planes.

6) Use a triple integral to find the volume of the solid bounded by y = 4 - x^2, z = 0, and z = y.

7) Use a triple integral to find the volume of the solid bounded by x^2 + y^2 + z^3 = 8 and z = 1.

8) Use a triple integral to find the volume of the solid bounded by the cylinders y = z^2 and y = 2 - z^2 and the planes x = 1 and x = -2.

9) Change the order of integration of the triple integral Int[Int[Int(f(x,y,z) dz, 0, 1-2x) dy, 0, 1-4x^2] dx, 0, 1/2] to dy dx dz.

10) Find the volume of the region between the two elliptic paraboloids z = x^2/4 + y^2 - 9 and z = -x^2/4 - y^2 + 9.

11) Use triple integration to find the volume of the following solids.

• A sphere of radius r.
• An ellipsoid with equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 (a,b,c > 0).

12)** Use the following steps to help to solve the integral I = Int[Int[Int(sin((pi - z)^3) dz) dy] dx] over V where D is the region bounded below by the xy-plane, above by the plane x = z, and laterally by the planes x = y and y = pi.

• Change the order of integration to show that Int[Int (f(u) du,0,v) dv, 0, x] = Int( (x-u)*f(u) du, 0, x).
• Change the order of integration to show that Int[Int[Int(f(w) dw, 0, u) du, 0, v] dv, 0, x] = 1/2*Int( (x-w)^2*f(w) dw, 0, x).
• Use the two above steps to evalutate I.

13) Evaluate the 4-tuple (quadruple) integral of e^(x +2y - z - w) with respect to w, z, y, x in that order over H. H is the four-dimensional region bounded by the hyperplane x + y + z + w = 4 and the coordinate spaces x =0, y = 0, z = 0 and w = 0 where x,y,z,w >= 0.

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Section 12.6

1) Find the centroid for a lamina with delta = 4 over the region bounded by the curve y = sqrt(x) and the line x = 9 in the first quadrant.

2) The part of the spherical solid with density delta = 2 described by x^2 + y^2 + z^2 <= 9, x,y,z >= 0.

3) Use double integration to find the center of mass when delta(x,y) = k*(x^2 + y^2) over the region x^2 + y^2 <= a^2, y >= 0.

4) Use double integration to find the center of mass when delta(x,y) = y over the region y = e^-x, x = 0, x = 3, y = 0.

5) Find I_x, the moment of inertia about the x-axis, of the lamina that covers the region bounded by the graph of y = 3 - x^2 and the x-axis, and with density delta(x,y) = x^2*y^2.

6) Find the center of mass of the cardioid r =1 + sin(theta) if the density at each point (r,theta) is delta(r,theta) = r.

7) Show that a homogenous lamina of mass m that covers the circular region x^2 + y^2 = a^2 will have moment of inertia ma^2/4 with respect to both the x- and y-axes. What is the moment of inertia with respect to the z-axis?

8) Find the center of mass of the tetrahedron in the first octant bounded by the plane x/a + y/b + z/c where a,b,c > 0. Assume delta = x.

9) Suppose the joint probability density function for the random variables X and Y is f(x,y) = {4e^(-2x)*e^(-y) if x,y >= 0, 0 otherwise.} Find the probability that X + Y <= 1.

10) Suppose X measures the time (in minutes) that a person stands in line at a certain bank and Y, the duration (in minutes) of a routine transaction at the teller's window. You arrive at the bank to deposit a check. If the joint probability function for X and Y is modeled by f(x,y) = { 1/16*e^(-x/4)*e^(-y/2) if x,y >= 0, 0 otherwise} Find the probability that you complete your business at the bank within 10 minutes.

11) Find the average value of the function f(x,y,z) = x + 2y + 3z over the solid region S bounded by the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1).

12) The radius of gyration for revolving a region R ,with mass m, about an axis of rotation ,with moment of inertia I, is d = sqrt(I/m). Find the radius of gyration about the x-axis of the semicircular region x^2 + y^2 <= a, y >= 0 given that the density at (x,y) is directly proportional to the distance of the point from the x-axis.

13) A solid has the shape of a rectangular parallelepiped given by -a <= x <= a, -b <= y <= b, -c <= z <= c and its density is delta(x,y,z) = x^2 * y^2 * z^2.

• Guess the location of the center of mass and the value of the moment of inertia about the z-axis.
• Check your guess by direct computation.

14)** Prove the following area theorem of Pappus: Let C be a curve of length L in the plane. The the surface obtained by rotating C about the axis L in the plane has area 2pi*Lh, where h is the distance from the centroid of C to the axis of rotation.

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Section 12.7

1) Convert the point (sqrt(2), 2, -sqrt(3)) from rectangular coordinates to both cylindrical and spherical coordinates.

2) Convert the point (-3, 2pi/3, 3) from cylindrical coordinates to both rectangular and spherical coordinates.

3) Convert the point (pi,pi,pi) from spherical coordinates to both rectangular and cylindrical coordinates.

4) Convert the equation z = x^2 + y^2 to cylindrical coordinates.

5) Convert the equation 3x^2 + 3y^2 + 3z^2 = 1 to spherical coordinates.

6) Convert the equation rho = sin(theta)*cos(phi) to rectangular coordinates.

7) Evaluate the iterated integral Int[ Int[ Int(rho^2*sin(phi) d(rho), 0, cos(phi)) d(theta),0, pi/4] d(phi), 0, pi/2].

8) Evaluate the iterated integral Int[ Int[ Int( r dz, 0, 4cos(theta)) dr, 0, sin(theta)] d(theta), 0, pi/2].

9) Use cylindrical coordinates to compute the triple integral of z(x^2 + y^2)^(-1/2) with respect to V over D where D is the solid bounded by the plane z = 2 and below by the surface 4z = x^2 + y^2.

10) Evaluate the triple integral of sqrt(x^2 + y^2 + z^2) with respect to V over R where R is the region defined by x^2 + y^2 + z^2 <= 5.

11) Find the volume of the region D where D is the region defined by the paraboloid z = 1 - 9(x^2 + y^2) and the xy-plane.

12) Find the volume of the region D where D is the region defined as the intersection of the solid sphere x^2 + y^2 + z^2 <= 16 and the solid cylinder x^2 + y^2 <= 4.

13) Let D be a homogenous solid (with density delta) that has the shape of a right circular cylinder with height and radius r. Use cylindrical coordinates to find the moment of inertia of S about its axis of symmetry.

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Section 12.8

1) Find the Jacobian d(x,y)/d(u,v) (For this section, d denotes the partial symbol) when x = 2u - v and y = 4u - 3v.

2) Find the Jacobian d(x,y,z)/d(u,v,w)  when x = 2u - v, y = 2v + 2w, z = v - w.

3) Find the Jacobian d(x,y)/d(u,v)  when u = 3x + y and v = -x + y.

4) Find the Jacobian d(x,y)/d(u,v) when u = x/(x^2 + y^2) and v = y/(x^2 + y^2).

5) Let R be the parallelogram with vertices (0,0), (1,4), (4,6), (4,2). Sketch and describe the corresponding region after the transformation u = x^2, v = x+ y.

6) Suppose the uv-plane is mapped onto the xy-plane by the equations x = 2uv, y = u^2 - v^2. Express dx dy in terms of du dv.

7) The region R bounded by the parabolas y = x^2, y = 4x^2, y = sqrt(x), and y = 1/2sqrt(x).

8) Let D be the region in the xy-plane which is bounded by the coordinate axes and the line x + y = 1. Use the change of variables u = x + y and v = x - y to compute the double integral of (x-y)*e^(x^2 + y^2) over D with respect to A

9) Under the transformation u = 1/5(2x + y) and v = 1/5(x - 2y) the region D which is a square in the xy-plane with vertices (0,0), (1,-2) is mapped onto a square in the uv-plane. Use this information to find the integral of cos(2x + y)*sin(x - 2y) with respect to V over D.

10) Under the change of variables x = s^2 - t^2, y = 2st, the quarter circular in the st-plane given by s^2 + t^2 <= 1, s >= 0, t >= 0 is mapped onto a certain region D* of the xy-plane. Find the integral of 1/(sqrt(x^2 + y^2) over D* with respect to V.

11)** Use a change of variables to find the moment of inertia about the z-axis of the solid ellipsoid with density 1 and equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1.

12) A rotation of the xy-plane through the fixed angle theta is given by x = u cos(theta) - v sin(theta), y = u sin(theta) + v cos(theta).

• Compute the Jacobian of this transformation.
• Let E denote the ellipse x^2 + xy + y^2 = 9. Use a rotation of pi/4 to obtain an integral which is equivalent to the double integral of y with respect to V over E.
• Evaluate the integral found in the previous step.

13) Find the Jacobian of the cylindrical coordinate transformation x = r cos(theta), y = r sin(theta), z = z.

14)** Let T: x = x(u,v), y = y(u,v) be a one-to-one transformation on a set D so that T^-1 has the form u = u(x,y), v = (x,y). Use the multiplicative property of determinants along with the chain rule, to show that d(x,y)/d(u,v) * d(u,v)/d(x,y) = 1. You may assume that all needed partial derivatives exist.

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Section 13.1

1) Find div F and curl F when F(x,y) = i + (x^2 + y^2)k.

2) Find div F and curl F when F(x,y) = (sin x)i + (cos x)j + k at the point P = (pi, pi/4, 0).

3) Find div F and curl F when F(x,y) = (e^y cos x)i + (e^y sin x)j + k at the point P = (3,-1,-2).

4) Find div F and curl F when F = xi - yj.

5) Find div F and curl F when F = axi + byj + czk, where a,b,c are real numbers.

6) Find div F and curl F when F = (xi + yj + zk)/sqrt(x^2 + y^2 + z^2).

7) Is the function r(x,y,z) = xyz harmonic?

8) Find div F, given that F = grad(f), where f(x,y,z) = x^2*y^3*z.

9) If F(x,y,z) = i + 3xj + 2yk and G(x,y,z) = 2xi - 3yk +5 zk, then find div(F X G).

10) If F(x,y) = u(x,y)i + v(x,y)j, show that curl F = 0 if and only if du/dy = dv/dx (Here d denotes the partial symbol).

11) Which (if any) of the following is the same as div(F X G) for all vector fields F and G.

•  (div F)(div G)
• (curl F) dot G - F dot (curl G)
• F(div G) + (div F)G
• (curl F) dot G + F dot (curl G)

12) Prove that the curl of the gradient of a function is always 0.

13) Let R = <x,y,z> and r = ||R|| = sqrt(x^2 + y^2 + z^2). If curl R = 0 then what is div R?

14)** Let F = <x^2y, yz^2, zy^2>. Either find a vector field G such that F = curl G, or show that no such G exists.

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Section 13.2

1) Evaluate Int[ (2x - 3y) ds, C] where C is defined by x = sin t, y = cos t, 0 <= t <= pi. ( Int[ f(x,y) ds, C] is the line integral of f(x,y) over C).

2) Evaluate Int[-y dx + 3y dy, C] where C is defined by y^2 = x from the point (1,1) to the point (9,3).

3) Evaluate Int[ (y-x) dx + (x^2*y) dy, C] where C is the quarter-circle x^2 + y^2 = 4 from (0,2) to (2,0).

4) Consider Int[(y^2 - x^2) dx - x dy, C]. Evaluate this integral for the following curves C:

• C is the arc of the parabola y = x^2 from (0,0) to (2,4).
• C is the segment of the line y = 2x for 0 <= x <= 2.

5) Evaluate Int[(x^2 - y^2)dx + x dy, C] where C is the circular path given by x = 2 cos(theta), y = 2sin(theta), 0 <= theta <= 2pi.

6) Evaluate Int[ F dot dR, C] where F = (5x + y)i + xj and C is the vertical line fron (0,0) to (0,2) followed by the horizontal line from (0,2) to (2,2)

7) Consider Int[ -y dx + x dy + xz dz, C]. Evaluate this integral for the following curves C:

• C is the helix defined by x = cos t, y = sin t, z = t for 0 <= t <= 2pi.
• C is the unit circle x^2 + y^2 = 1, z = 0, traversed once counterclockwise as viewed from above.

8) Evaluate Int[F dot dR, C], where F = (y - 2z) i + xj - 2xyk and C is the path given by R(t) = ti + t^2j - k for 1 <= t <= 2.

9) Evaluate Int[ye^(xz) ds, C], where C is the line segment from (0,0,0) to (2,1,3).

10) Evaluate Int[F dot T ds] where F = -3yi + 3xj + 3xk and C is the straight line segment from (0,0,1) to (1,1,1).

11) Evaluate Int[ (x^2 + xy + y^2)/z^2 ds, C], where C is the path given by R(t) = (cos t)i + (sin t)j - k.

12) Evaluate Int[ (dx + dy)/(|x| + |y|), C] where C is the square |x| + |y| = 1, traversed once clockwise.

13) A force acting on a point mass located at (x,y) is given by F = yi + 2xj. Find the work done by this force as the point mass moves along a straight line from (1,0) to (0,1).

14)** Find the center of mass of a wire in the shape of the helix x = 3 sin t,  y = 3 cos t, z = 2t for 0 <= t <= pi and the density delta(x,y,z) as follows:

• delta(x,y,z) = z
• delta(x,y,z) = x

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Section 13.3

1) Determine if the vector field F= (xe^(xy) sin y)i + (e^(xy)cos xy + y)j is conservative, if it is then find a scalar potential.

2) Determine if the vector field F = (y- x^2)i + (2x + y^2)j is conservative, if it is find a scalar potential

3) Consider the integral Int[ 2x^2y dx + x^3 dy, C]. Evaluate it on each of the following curves C:

• The semicircle x = sqrt(1-y) from -1 <= y <= 1.
• The line segment from (0,-1) to (1,1) and then the line segment from (1,1) to (0,1).
• The square with vertices (1,1), (-1,1), (-1,-1), (1,-1) traversed once clockwise.

4) Show that F = e^-yi - xe^-yj is conservative and find a scalar potential f for F. Evaluate the line integral  Int[ F dot dR, C], where C is any smooth path connecting (0,0) and (1,1).

5) Show that the vector field F = (y sin z)i + (x sin z + 2y)j + (xy cos z)k is a conservative vector field and find the scalar potential f for F.

6) Show that the vector field F= <3x^2*y^2*z, 2x^3*yz, x^3*y^2 - e^-z> is conservative and evaluate Int[ F dot dR, C], where C is any smooth path connecting (1,0,-1) to (0,-1,1).

7) Show that the vector field F = <2xz^3 - e^(-xy)y sin z, -xe^(-xy) sin z, 3x^2*z^2 + e^(-xy)cos z> is conservative and evaluate Int[ F fot dR, C], where C is any smooth path connecting (1,0,-1) to (0,-1,1).

8) Verify that the integral Int[(xy cos (xy) + sin (xy))dx + (x^2 cos(xy)) dy, C] is independent of path and evaluate it where C is any path from (0, pi/18) to (1, pi/6).

9) Evaluate the line integral Int[ (xy^2 i + x^2yj) dot dR, C] Where C is any path from (4,1) to (0,0).

10)** Find a function g so that g(x)F(x,y) is conservative where F(x,y) = (x^2 + y^2 + x)i + xyj.

11) Consider the line integral I = Int[(-y/x^2 + 1/x)dx + (1/x)dy, C].

• ** Over what region in the xy-plane will the line integral I be independent of path?
• Evaluate the line integral I if C is defined by R(t) = (cos^3(t))i + (sin 3t)j for 0 <= t <= pi/3.

12) Let F(x,y) = (-yi + xj)/(x^2 + y^2).

• Compute Int[F dot dR, C1] where C1 is the upper semicircle y = sqrt(1-x^2) traversed counterclockwise.
• Compute Int[F dot dR, C2] where C2 is the lower semicircle y = -sqrt(1-x^2) traversed counterclockwise.
• **Show that if F = Mi + Nj then d/dy(-y/(x^2 + y^2)) = d/dx(x/(x^2 + y^2)) but that F is not conservative on the unit circle x^2 + y^2 = 1. (d denotes the partial symbol here)

13) Show that if the vector field F(x,y,z) = M(x,y,z)i + N(x,y,z)j + P(x,y,z)k is conservative then dP/dy = dN/dz, dM/dz = dP/dz, dN/dx = dM/dy. (d denotes the partial symbol here)

14) Let f and g be differentiable functions of one variable. Show that the vector field F = [f(x) + y]i + [g(y) + x]j is conservative and find the corresponding potential function.

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Section 13.4

1) Use Green's Theorem to evaluate Int[ y^3 dx - x^3 dy, C] where C is parameterized by x = cos(theta), y = sin(theta) and 0 <= theta <= 2pi. Check your answer by computing the line integral without Green's Theorem.

2) Use Green's Theorem to evaluate Int[4y dx - 3x dy, C] where C is the ellipse 2x^2 + y^2 = 4. Parameterize C to find the value of the integral without using Green's Theorem to check your answer.

3) Use Green's Theorem to evaluate Int[ e^x dx - sin x dy, C] where C is the square with vertices (0,0), (1,0), (1,1), (0,1) traversed counterclockwise.

4) Use Green's Theorem to evaluate the integral Int[ (x + y)dx - (3x - 2y)dy, C] where C is the trapezoid with vertices (0,0), (0,4), (2,4), (3,0) traversed clockwise.

5) Use Green's Theorem to evaluate the integral Int[2x arctan(y) dx - x^2*y^2.(1 + y^2) dy, C] where C is the square with vertices (0,0), (3,0), (3,3), (0,3) traversed clockwise.

6) Find the work done when an object moves in the force field F(x,y) - 2y^2i + 3x^2j counterclockwise around the circular path x^2 + y^2 = 4.

7) Using a line integral, find the area enclosed in the triangle with vertices (0,0), (1,2), (0,3). Check your answer using a geometric formula.

8) Using a line integral, find the area enclosed in the semicircle y = sqrt(9 - x^2). Check your answer using a geometric formula.

9) Evaluate the integral Int[ -2y dx + 3x dy, C] where C is the cardioid r = 1 + sin(theta) traversed counterclockwise.

10)** Using a line integral and polar transformation formulas, show that the area formula in polar coordinates is A = 1/2 Int[ r^2 d(theta), (theta1), (theta2)] = 1/2 Int[(g(theta))^2 d(theta), (theta1), (theta2)].

11) Evaluate Int[ (x dx + y dy)/(x^2 + y^2), C] where C is any smooth Jordan curve...

• ... not enclosing the origin traversed counterclockwise.
• ...enclosing the origin traversed counterclockwise.

12) Evaluate Int[ dz/dn ds, C] where z(x,y) = 2x^2 + 3y^2, and C is the circular path x^2 + y^2 = 16 traversed counterclockwise. (Here d denotes the partial symbol).

13) Evaluate Int[ x*dx/dn ds, C] where C is the bounday of the unit square 0 <= x,y <= 1 traversed counterclockwise. (Here d denotes the partial symbol)

14)** Suppose f is a harmonic function over D and D is a simply connected region enclosed by the Jordan curve C, show that the double integral of (f_x^2 + f_y^2) over D is equal to Int[ f*df/dn ds, C]. (Here d denotes the partial symbol.)

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Section 13.5

1) Evaluate the surface integral Int[Int[2xy dS, S]] where S is the surface described by z = 10, and x^2/4 + y^2 <= 1.

2) Evaluate the surface integral Int[Int[(x^2 + y^2) dS, S]] where S is the surface described by z = 4 - x, 0 <= x,y <= 2.

3) Evaluate the surface integral Int[Int[(x^2 + y^2) dS, S]] where S is the surface described by z = xy, x^2 + y^2 <= 4, x,y >= 0.

4) Let S be the hemisphere x^2 + y^2 + z^2 = 4 with z >= 0. Evaluate Int[Int[(z^2) dS,S]].

5) Let S be the hemisphere x^2 + y^2 + z^2 = 4 with z >= 0. Evaluate Int[Int[(x^2 + y^2) dS,S]].

6) Let S be the portion of the paraboloid z = x^2 + y^2 where z <= 4. Evaluate Int[Int[1/sqrt(1 + 4z) dS, S]].

7) Evaluate Int[Int[ F dot N dS, S]], where F = xi + 2yj + zk, S is the surface of the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, z = 1. N is the outward directed normal field.

8) Evaluate Int[Int[ F dot N dS, S]], where F = 2xi - 3yj, and S is the part of the hemisphere given by x^2 + y^2 + z^2 = 5 for z >= 1., N is the outward directed normal field.

9) Evaluate Int[Int[ (3x - y + 2z) dS,S]], where S is given by the parameterization R(u,v) = ui + uj - vk, 0 <= u <= 1, 1 <= v <= 2.

10) Evaluate Int[Int[ (arctan x + y - z^2) dS, S]] where S is given by the parameterization R(u,v) = ui + v^2j - vk.

11) Evaluate Int[Int[F dot N dS, S]], where F = xi + yj + z^4k and S is the parametric surface x = u cos v, y = u sin v, z =u for 0<= u <= 2, 0 <= v <= 2pi.

12) Find the mass of the homogenous lamina that has the shape of S where S is the surface z = 1 - x^2 - y^2, with z >= 0 and delta = x^2 + y^2 + z^2.

13)** Show that the moment of inertia of a conical shell about its axis is 1/2ma^2 where m is the mas and a is the radius of the cone. Assume delta(x,y,z) = 1.

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Section 13.6

1) Verify Stokes' theorem given F = 2xyi + z^2k, and the surface S being the portion of the paraboloid y = x^2 + z^2 with y <= 4.

2) Use Stokes' theorem to evaluate the line integral Int[x^3y^2 dx + dy + z^2 dz,C] where C is the circle x^2 + y^2 = 1 in the plane z = 1 traversed counterclockwise when viewed from the top.

3) Use Stokes' theorem to evaluate the line integral Int[y dx - 2x dy + z dz, C] where C is the intersection of the surface z = x^2 + y^2 and the plane x + y + z = 1 considered counterclockwise when viewed from above the origin.

4) Use Stokes' theorem to evaluate the line integral Int[y dx + z dy + y dz, C] where C is the intersection of the sphere x^2 + y^2 + z^2 = 4 and the plane x + y + z = 0, traversed counterclockwise when viewed from above the origin.

5) Evaluate Int[ F dot dR, C] where F = <x - z, y - x, z - y> and C is the boundary of the triangular region with vertices (12,0,0), (0,3,0), (0,0,12) traversed counterclockwise as viewed from above.

6) Use Stokes' theorem to evaluate Int[Int[ curl F dot N dS,S]] where F = yi + zj + xk and S is the part of the plane x + y + z = 1 that lies in the first octant.

7) Use Stokes' theorem to evaluate Int[Int[ curl F dot N dS,S]] where F = <x arctan(e^-x), y ln(1 + y^(3/2)), ze^(-1/z) and S is the part of the sphere x ^2 + y^2 + z^2 = 9 that lies inside of the cone z = sqrt(2x^2 + 2y^2

8) Use Stokes' theorem to evaluate the line integral Int[ (1 + y)z dx + (1 + z)x dy + (1 + x)y dz, C] for any closed path C in the plane 2x - 3y + z = 1.

9) The vector field V represents the velocity of a fluid flow, find the circulation Int[V dot dR, C] assuming a counterclockwise orientation when viewed from above where V = yi + ln(x^2 + y^2)j + (x + y)k, and C is the triangle with vertices (0,0,0), (1,0,0), (0,1,0).

10)** Let F = zi + xj + yk and suppose S is a smooth surface in R^3 whose boundary is given by x = 2 cos(theta), y = 3 sin(theta) and z = sin(theta) where 0 <= theta <= 2pi. Use Stokes' theorem to evaluate Int[Int[ curl F dot N dS, S]].

11)** Faraday's law of electromagnetism says that if E is the electric intensity vector in a system, then Int[ E dot dR, C] = -d(phi)/dt (Here d denotes the partial symbol) around any closed curve C, where t is time and phi is the total magnetic flux directed outward through any surface S bounded by C. Given that phi = Int[Int[B dot N dS,S]] where B is the magnetic flux density show that curl E = -dB/dt. Note that it can be shown that d/dt*Int[Int[B dot N dS,S]] = Int[Int[dB/dt dot N dS,S].

12)**Suppose f and g are functions of x,y,z with continuous first- and second-order partial derivatives and C is a closed curve bounding the surface S. Use Stokes' Theorem to verify that Int[ f*grad(g) dot dR, C] = Int[Int[ grad(f) X grad(g)) dot N dS, S]].

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Section 13.7

1) Verify the divergence theorem for F = xzi + y^2j + 2zk where D is the ball x^2 + y^2 + z^2 <= 4. Assume N is the unit normal vector field pointing away from the origin.

2) Use the divergence theorem to evaluate the surface integral Int[Int[ F dot  N dS, S]] where F = xyzj, S is the cylinder x^2 + y^2 = 9 for 0 <= z <= 5 and N is the outward unit normal vector field.

3) Use Stokes' theorem to evaluate Int[Int[ curl F dot N dS,S]] where F = 2yi - zj + 3xk, S is the surface comprised of the five faces of the unit cube 0 <= x,y,z <= 1 where z = 0 is missing and N is the outward unit normal vector field.

4) Use Stokes' theorem to evaluate Int[Int[ curl F dot N dS,S]] where F = curl(yi + xj - zk), S is the hemisphere z = sqrt(4 - x^2 - y^2) together with the disk x^2 + y^2 <= 4 in the xy-plane and N is the outward unit normal vector field.

5) Use Stokes' theorem to evaluate Int[Int[ curl F dot N dS,S]] where F= xy^2i + yz^2j + x^2zk; S is the surface bounded above by the sphere rho = 2 and below by the cone phi = pi/4 (in spherical coordinates) and N is the outward unit normal vector field.

6) Suppose that S is a solid surface which encloses a solid region D. Let N be an outward unit normal vector to S

• **Show that the volume of D is given by V(D) = 1/3 Int[Int[(xi + yj + zk) dot N dS, S]].
•  Use the formula to find the volume of the hemisphere z = sqrt(R^2 - x^2 - y^2).

7) The moment of inertia about the z-axis of a solid D of constant density delta = a is given by Iz = Int[Int[Int a(x^2 + y^2) dV, T]]]. Express this integral as a surface integral over the surface S that bounds D.

8) Show that if g is harmonic in the region D, then Int[ Int[ dg/dn dS, S]] = 0 where the closed surface S is the boundary of D. (Here d denotes the partial symbol).

9) An electric change q located at the origin produces the electric field E = qR/(4*pi*epsilon*||R||^3). Where R = xi + yj + zk and epsilon is a physical constant called the electic permittivity.

• Show that Int[Int[E dot N dS, S]] = 0 if the closed surface does not enclose the origin. This is called Gauss's law.
• Show that Int[Int[E dot N dS, S]] = q/epsilon when the surface does enclose the origin, note that the divergence theorem does not apply directly to this case.

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