Use Stokes' Theorem to evaluate ??S curl F

To evaluate curl F ds, we need to first determine the function F. F is a function that takes thr

Use Stokes' Theorem to evaluate ??_S curl F · dS. F(x, y, z) = x^2y^3zi + sin xyzj + xyzk, S is the part of the cone y^2 = x^2 + z^2 that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.

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Transcript

00:01 In this problem we are asked to evaluate the double integral over the surface as curl of f, d s.

00:13 Here, f of x, y, z equals to x squared y cubed z i vector plus sign of x, y z j vector plus x, y z, j vector plus x, y z k vector, and the surface s is the region of the cone, y squared equals to x squared plus z squared that lies between y equals to 0 and y equals to 2 and the positive y axis.

00:54 So here we make use of the stokes theorem.

00:57 The stokes theorem states that the double integral over s curl of fds equals equals.

01:03 Equals to the integral over c fdr.

01:08 So here we need to parameterize c.

01:13 We can rewrite x as 2 times cos of t, y as 2 times sine of t and sorry z as 2 times sine of t and y as 2.

01:29 So we obtain r of t to be equal to 2 2 cos of t to 2 sine of t.

01:40 So now let us differentiate this.

01:43 We get dr to be equal to negative 2 sine of t 0 and positive 2 cost of t.

01:51 Now let us substitute this value of r in f.

01:56 So we get f of r of t to be equal to 4 times cos squared t times 8.

02:07 Times 2 times sine of t i vector plus sign of 8 times cos of t sine of t j vector plus 8 times cos of t times sine of t k vector so simplifying this we have f of t to be equal to 64 cost square t times sine of t, sign of 8 times cos of t sign of t, and 8 times cost of t, sign of t...

Question

Use Stokes' Theorem to evaluate ??_S curl F · dS. F(x, y, z) = x^2y^3zi + sin xyzj + xyzk, S is the part of the cone y^2 = x^2 + z^2 that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.

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