Evaluate F⃗· dr⃗ where F⃗(x, y, z) = (4yz cos(xyz), 4xz...

Evaluate $\int_C \vec{F} \cdot d\vec{r}$ where $\vec{F}(x, y, z) = (4yz \cos(xyz), 4xz \cos(xyz), 4xy \cos(xyz))$ and C is the line segment joining the point $(3, \frac{\pi}{2}, \frac{1}{3})$ to the point $(6, \frac{1}{6}, \frac{\pi}{6})$.

Transcript

00:01 Okay, so we want to evaluate this line integral, and the line is a straight line from this point to this point.

00:07 So i'm going to take t to be in the interval 0 -1, and then we need to parameterise x of t, y of t, and z of t, of course.

00:19 So z of t.

00:22 So if we do this, these are all straight lines, and they just need to connect between the correct points.

00:28 So z of t, i'm going to do this in the reverse order, because that's the easiest.

00:31 Z of t is just equal to t.

00:33 So when t is zero, z is zero, when t is one, z is one, and it's a straight line.

00:38 Y of t, we can just write as t plus five.

00:42 So when t is zero, y is five, when t is one, y is six.

00:46 And then x of t is going to be 2t minus 2.

00:52 So when t is equal to 0, we just have minus 2.

00:56 And then when t is equal to 1, we have 2 minus 2, which is 0, which is this endpoint.

01:02 Okay.

01:03 So this is our equation.

01:08 These are our parameterizations.

01:10 And we're just going to have ds is equal to dt, since this is a straight line.

01:15 So now we can do this integral.

01:17 This is going to be the integral from zero to one, since this is where our t varies.

01:23 And then we have x, which is 2t minus 2.

01:27 We have y, which is t plus 5.

01:29 And we have z squared, which is 2 .2.

01:31 And we have z squared, which is t squared and this is dt.

01:33 So now we just need to integrate.

01:35 So now we just have a load of algebra.

01:37 So let's expand these brackets first and see what we get.

01:40 We get 2 t squared plus 10 t minus 2.

01:45 This is a t.

01:46 T minus 10 times t squared d t.

01:52 Without writing another line, 10t minus 2t is 8t.

01:56 So let's just change this to plus 8t.

02:01 And move this across...

Question

Evaluate $\int_C \vec{F} \cdot d\vec{r}$ where \(\vec{F}(x, y, z) = (4yz \cos(xyz), 4xz \cos(xyz), 4xy \cos(xyz))\) and C is the line segment joining the point \((3, \frac{\pi}{2}, \frac{1}{3})\) to the point \((6, \frac{1}{6}, \frac{\pi}{6})\).

Question

Evaluate $\int_C \vec{F} \cdot d\vec{r}$ where \(\vec{F}(x, y, z) = (4yz \cos(xyz), 4xz \cos(xyz), 4xy \cos(xyz))\) and C is the line segment joining the point \((3, \frac{\pi}{2}, \frac{1}{3})\) to the point \((6, \frac{1}{6}, \frac{\pi}{6})\).

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