Find the stream function for the vector field F =...

Find the stream function for the vector field $F = -\frac{49y^5}{49}\hat{i} - \frac{49x^6}{49}\hat{j}$ \\ If C is a curve from the point P = (2, 3) to the point Q = (2, 4), then the flux across C is given by:

Transcript

00:01 Okay, so let's write out what the vector field is.

00:04 Well, the vector field f is just minus y to the fifth, x to the sixth.

00:18 Okay, and you want to find the stream function for this.

00:20 So what is the stream function? the stream function is, okay, there's different conventions, but stream function is the function such that if i take the first one, this should be, if i take the first one, this should be partial if i take, so psi is the stream function, psi stream function, then the first one should be partial psi over partial y, and the second one should be minus partial psi over partial x.

01:11 Okay, so this is, there's also a sign convention that sometimes the minus is written in the other place.

01:19 Actually, for us, maybe it's easier to write minus in the other place.

01:22 Let's do this convention.

01:24 Okay, so let's do this convention.

01:29 So how do we find the psi? well, we just integrate.

01:32 So let's integrate the first one with respect to y.

01:35 So minus partial psi over partial y means that we just integrate y to the fifth.

01:42 So minus of the first one, dy.

01:46 So we're doing a partial integration.

01:48 So what is this integration? this is y to the sixth over six plus, since we did indefinite, there's no limits.

01:59 We need to, and since we're only doing partial derivatives, partial derivatives don't see anything, any other functions of x, for example.

02:07 So you can say plus some g of x.

02:10 So this is the partial integration for the first one.

02:16 And you can see that if i take the partial derivative with respect to y of this quantity, you get exactly, and put a minus sign, you get exactly.

02:29 Okay, now let's do a partial integration of the second thing.

02:32 So it's x to the sixth dx.

02:35 So this is going to be x to the seventh over seven plus some h of y.

02:46 And then you need to try to match these two together...

Question

Find the stream function for the vector field $F = -\frac{49y^5}{49}\hat{i} - \frac{49x^6}{49}\hat{j}$ \\ If C is a curve from the point P = (2, 3) to the point Q = (2, 4), then the flux across C is given by:

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