Use Stokes' Theorem to evaluate curl F ·dS . F(x, y, z) = e^xy i + e^xz j + x^2z k , S is

step 1 All right, let's go ahead and solve this problem. So we're, as usual, we are asked to evaluate t

Use Stokes' Theorem to evaluate curl F ·dS . F(x, y, z) =...

Use Stokes' Theorem to evaluate $ \iint_S \text{curl} \textbf{F} \cdot d\textbf{S} $. $ \textbf{F}(x, y, z) = e^{xy} \, \textbf{i} + e^{xz} \, \textbf{j} + x^2z \, \textbf{k} $, $ S $ is the half of the ellipsoid $ 4x^2 + y^2 + 4z^2 = 4 $ that lies to the right of the $ xz $-plane, oriented in the direction of the positive $ y $-axis

Use Stokes' Theorem to evaluate $ \iint_S \text{curl} \textbf{F} \cdot d\textbf{S} $.

$ \textbf{F}(x, y, z) = e^{xy} \, \textbf{i} + e^{xz} \, \textbf{j} + x^2z \, \textbf{k} $,
$ S $ is the half of the ellipsoid $ 4x^2 + y^2 + 4z^2 = 4 $ that lies to the right of the $ xz $-plane, oriented in the direction of the positive $ y $-axis

Transcript

00:01 All right, let's go ahead and solve this problem.

00:04 So we're, as usual, we are asked to evaluate this double integral of curlf.

00:12 F, d .s.

00:13 Okay? it's probably not the best idea to do this straightforwardly because, as you can see, f is already quite involving.

00:23 You have e, you have x, y, x, z, a bunch of cross terms.

00:28 So the curl of that expression is not going to look anything simple.

00:33 And the surface is an ellipsoid, which makes things a little bit even more complicated.

00:40 However, using stokes theorem, we can simplify this problem significantly.

00:44 So let's take a look at what it says.

00:48 First, we're going to be able to say that this is equal to a single integral of through c of f.

00:56 Dot d r so it's a line integral okay so we want to be able to figure out what c is as a projection and then we want to basically parametize that so we can do the calculation all right so the ellipsoid is right here you can find out that y is from zero to four and the projection onto the x z plane is simply a circle so c is a a circle x squared plus z squared is equal to one okay so because of that i would like to choose the parameterization x equals to cosine theta we know that y is equal to zero and z is equal to sine theta.

01:56 Okay.

01:57 Now, where we take a look at the f in terms of the parameterization, we know that y is equal to zero.

02:12 So e to the xy is simply equal to e to the zero, or in other words, one.

02:20 X, z is e to the cosine theta, sine theta.

02:26 It's a little bit alarming, but don't worry.

02:34 And x squared times z is equal to cosine squared times sine.

02:45 Okay, so let's calculate what f...