If your solution to stated problem does not match the given solution, you should self-critique per instructions at

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Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

At the end of this document, after the qa problems (which provide you with questions and solutions), there is a series of Questions, Problems and Exercises. 

The del operation can be notated as d_x i + d_y j, where d_x means 'take the partial derivative with respect to x' and d_y means 'take the partial derivative with respect to y'.  Thus for example d_x f means 'take the partial derivative with respect to x of f', so that using f_x for the derivative with respect to x we have d_x f = f_x.

The del operator has already been seen in connection with the gradient.

The del operator is used to define gradient, divergence, and curl.  Chapter 13 explores the implications of these definitions.  In Section 13.1 we simply learn to perform the operations, and learn a few basic properties.

If f is a scalar function then del f = (d_x i + d_y j ) f = d_x f i + d_y f j =  f_x i + f_y j..  This is the gradient of f.

If F(x, y) = F_1(x, y) i + F_2(x, y) j, then the divergence of the function F is del dot F = (d_x i + d_y j ) dot (F_1 i + F_2 j ) = d_x F_1 + d_y F_2 = F_1_x + F_2_y.

The divergence and gradient can be defined for functions of 2 variables.

The 'curl' can be defined only for a function of 3 variables. 

If F(x, y) = F_1(x, y) i + F_2(x, y) j + F_3(x, y) k then the curl of F is

curl F = del X F = (d_x i + d_y j + d_z k) X (F_1 i + F_2 j + F_3 k) = (F_3_y - F_2_z) i - (F_3_x - F_1_z) j + (F_2_x - F_1_y) k

(this is easier to see in determinant notation).

Question:  Find div F and curl F when F(x,y) = x y i + (x^2 + y^2) j.

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where (x y)_x represents the partial derivative of xy with respect to x, and (x^2 + y^2)_y represents the partial derivative of x^2 + y^2 with respect to y.

The curl is defined only for a function of 3 variables, so we express our function as

F(x,y) = x y i + (x^2 + y^2) j + 0 k

This is of the form F_1 i + F_2 j + F_3 k for F_1 = x y, F_2 = x^2 + y^2 and F_3 = 0.

The curl is

curl F = del X F = (d_x i + d_y j + d_z k) X (F_1 i + F_2 j + F_3 k)

= (F_3_y - F_2_z) i - (F_3_x - F_1_z) j + (F_2_x - F_1_y) k

= (0 - 0) i - (0 - 0) j + (2 x - x) k

= x k.

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Additional questions will be added soon.

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