Consider the force field F F(x,y, z) = [2x - 2x sin (x^2

Consider the force field F F(x,y, z) = [2x - 2x sin (x^2 - y - z)] i + [sin (x^2 - y - z) + 1] j + [sin (x^2 - y - z) + 1] k. (a) Show that F is conservative (include the full derivation of the result). (b) Find a potential function f of F. (c) Calculate the work done by the force F in moving an object from point (0, 0, 0) to point (0, ?/2, ?).

Transcript

00:01 Consider the following vector field f.

00:04 In our first question, question a, we want to show that f is conservative.

00:09 So a function, a vector field f is conservative if its curl is equal to 0.

00:19 So first let's define the curl.

00:21 The curl is defined as the partial derivative of fz with respect to y minus the partial derivative of fy with respect to z in the i component plus the partial derivative of fx with respect to z minus the partial derivative of fz with respect to x in the j component plus the partial derivative of fy with respect to x minus the partial derivative of fz with respect to y in the k component.

01:10 So let's calculate each of these six partial derivatives.

01:21 Starting with this first bracket, let's look at fz and differentiate with respect to y assuming that the other variables are constant and we obtain minus the sine of x squared minus y minus z.

01:43 Now let's look at fy and differentiate partially with respect to z and we obtain plus sine x squared minus y minus z which is equal to 0.

01:57 So we've just shown that this first bracket is in fact equal to 0.

02:06 Next let's calculate the y component of our curl.

02:14 Let's look at fx and differentiate partially with respect to z and we obtain 2x times the sine of x squared minus y minus z.

02:27 Next let's look at fy and differentiate with respect to x and we obtain 2x times the sine of x squared minus y minus z which will also vanish.

02:47 Now let's evaluate our z component, this bracket.

02:54 Starting with fy, the derivative of fy with respect to z will give us minus the sine of x squared minus y minus z minus the derivative of the z component with respect to y so minus, or rather plus now, sine x squared minus y minus z which will also vanish.

03:28 So we've just shown that the curl of f is in fact equal to 0 implying that f is conservative.

03:45 Now if f is conservative, this means that f can write as the gradient of some potential function, say small caps f.

03:57 And now the goal is to find the functional form of this potential function.

04:05 And we do so by developing this equation.

04:09 We have that the x component of our potential function will be equal to the x component of our vector field which is 2x times the sine of x squared minus y minus z.

04:31 We can draw a similar equation by looking at the y component.

04:34 So we have that fy will be equal to the sine of x squared minus y minus z plus 1.

04:44 And we have that fz will be equal to the sine of x squared minus y minus z plus 1.

05:05 And now we can integrate each of these expressions to solve for our scalar function.

05:16 So integrating this line to solve for f, we obtain x cubed minus, or rather plus, the cos of x squared minus y minus z...

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