1. Evaluate the work done by the vector field F(x, y) =...

1. Evaluate the work done by the vector field \overrightarrow{F}(x, y) = \left(5x^3 - 4x + 2, 4y^2 + 6 + 2x\right)\\ over the curve C: \overrightarrow{r}(t) = \left(A\cos(t), 6B\sin(t)\right), 0 \le t \le 2\pi.

Transcript

00:01 In this problem, we need to find out the work done here.

00:02 F is of x, y, z is given to us as yz times e to the power of xy plus z square i cap plus zx e to the power of xy plus 3y square z j cap plus e to the power of xy plus 2xz plus yq plus 1 k cap.

00:22 There are other equalities for this vector field.

00:25 That is, 2f divided by 2x is equivalent to yz e to the power of xy plus z square.

00:31 Let this be considered as equation 1.

00:32 Then we have 2f divided by 2y which is equivalent to xz e to the power of xy plus 3y square z.

00:39 Let this be considered as equation 2.

00:41 Next, we have 2f by 2z which is equivalent to e to the power of xy plus 2xz plus yq plus 1.

00:48 Let this be considered as equation 3.

00:49 Now, let us integrate the equation 1 with respect to x.

00:56 We obtain that f of x, y, z is equivalent to integral of yz times e to the power of xy plus z square dx which is equivalent to yz e to the power of xy divided by y plus z square x plus g of y, z.

01:12 Let us consider this to be equation 4.

01:13 Now, y and y get cancelled in this case and that is a simplified form.

01:16 Now, the constant of integration for this integration will be function of both y and z.

01:19 So, the constant of integration for this function will be function of both y and z.

01:24 That is g of y, z.

01:26 Now, we can differentiate 4.

01:27 Let us differentiate 4 with respect to y and then we can set it equivalent to 2.

01:35 Therefore, upon differentiating, we could get z x e to the power of xy plus g to the power of yz is equivalent to xz e to the power of xy plus 3y square z.

01:46 So, we differentiated and then we had set it equal to the equation 2.

01:53 So, upon solving this, we get the value that is z x e to the power of xy gets cancelled on both the sides.

01:57 So, we have just remained with 3y square z.

01:59 Now, integrating this equation with respect to y, we get g of y, z is equal to integral of theta y square z dy which is y cube z plus h of z.

02:15 Let us consider this to be equation 5...

Question

1. Evaluate the work done by the vector field \overrightarrow{F}(x, y) = \left(5x^3 - 4x + 2, 4y^2 + 6 + 2x\right)\\ over the curve C: \overrightarrow{r}(t) = \left(A\cos(t), 6B\sin(t)\right), 0 \le t \le 2\pi.

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