A vector force field 𝐅 is defined in Cartesian coordinates by 𝐅=F0[((y^3)/(3 a^3)+(y)/(a) e^x y

Name: Problem 28, Chapter 11: Mathematical Methods for Physics and Engineering
Uploaded: 2022-02-05T17:13:46-08:00
Duration: 17 min 8 s
Channel: Linda Winkler
Description: A vector force field 𝐅 is defined in Cartesian coordinates by 𝐅=F0[((y^3)/(3 a^3)+(y)/(a) e^x y / a^2+1) 𝐢+((x y^2)/(a^3)+(x+y)/(a) e^x y / a^2) 𝐣+(z)/(a) e^x y / a^2 𝐤] Use Stokes' theorem to calculate ∮L 𝐅 ·d 𝐫 where L is the perimeter of the rectangle A B C D given by A=(0,1,0), B=(1,1,0), C=(1,3,0) and D=(0,3,0)

step 1 Here we're going to look at Stokes theorem. So we are going to look at a situation where we have

A vector force field 𝐅 is defined in Cartesian coordinates...

Key Concepts

Stokes' Theorem

Stokes' theorem is a fundamental result in vector calculus which relates a line integral around a closed loop (or boundary) to a surface integral over the surface bounded by that loop. Essentially, it equates the circulation of a vector field along a closed curve to the total 'twist' (or curl) of the field over the enclosed surface. This theorem is especially useful when the direct computation of the line integral is difficult, because the surface integral of the curl may be simpler to evaluate.

Line Integral of a Vector Field

A line integral in the context of a vector field involves integrating the dot product of the vector field and an infinitesimal displacement vector along a curve. When taken over a closed loop, the line integral can measure physical quantities such as work done or the circulation of the field. In applying Stokes' theorem, the line integral around the boundary is converted into a surface integral of the curl of the vector field over the area enclosed by the boundary.

Curl of a Vector Field

The curl of a vector field quantifies the amount of rotation or 'circulation density' at each point in the field. It is a vector field itself, describing how and where the original field 'curls' or rotates. In the context of Stokes' theorem, the curl is integrated over a surface to assess the cumulative rotational effect, thereby providing a link between the local rotational behavior and the overall circulation along the boundary.

Surface Orientation and Boundary Consistency

For Stokes' theorem to apply correctly, the orientiation of the chosen surface and its boundary curve must be consistent, typically following the right-hand rule. This means that if the fingers of the right hand point along the direction of the curve, the thumb should point in the direction of the surface's normal vector. Proper consideration of orientation is crucial in ensuring that the signs of the integrals are calculated correctly.

Transcript

00:01 Here we're going to look at stokes theorem.

00:04 So we are going to look at a situation where we have a vector field, we'll call it f.

00:13 And what stokes theorem tells us is that the curl of f inside a region, a closed region that has a closed perimeter, that the curl integrated over the area is equal to the line integral of f around the boundary.

00:38 And i like to think of stokes ' theorem as an example or a special situation of the fundamental theorem of calculus, that a derivative and an integral are oppositely related or inverse functions to each other.

00:57 And the derivative that we are looking at is the curl, and the integral is sort of canceling that curl if we think about the function along the boundary as being the contour integral, the line integral.

01:17 So here we have a fairly complex.

01:21 It's not complex, but it's a very intricate vector field.

01:28 Which is a function of x, y, and z with an exponential function thrown in.

01:33 And the region that we're interested in is a rectangle in the xy plane.

01:42 So here's the region of interest.

01:45 It goes between zero and one in the x direction and between one and three in the y direction.

01:53 And it circulates in a counterclockwise sense.

01:58 So let's just talk a little bit about which side of stokes theorem would be a little bit easier to negotiate in this case.

02:17 If we look at the line integral, f .dr, we can break it apart as fx, dx, plus fy, dy, d, plus fz d z okay and we see that there is no motion about the z direction so we can cancel that but what we would need to do is between a and b we have a d x with y equal to one or x equals 0 to 1 and then we have f y d y between between y equals 1 and 3 with x equals to 1.

03:37 And then we have fx d x with y equals 2 3 going from 1 to 0 and then 321 f y with x equals to 0.

03:59 So basically, the fx and the fy have lots and lots of parts to them, and that may be a little bit tricky to keep track of them.

04:12 So rather than evaluate all those different parts, what we're going to do is to take a look at the curl of f dotted into da and talk a little bit about that.

04:31 So use the curl to evaluate f .d .r.

04:40 Maybe we could go to all that work, and we should get agreement.

04:46 Da, in this case, it is in the xy plane.

04:55 And if we use the right -hand rule and look at the circulation, the direction is in the k direction.

05:03 Okay, so we need only to find one component of the curl.

05:22 And if you want to work that out, this is equal to the partial of x with respect to f y minus the partial with respect to y of f x.

05:41 So those partials need to be worked out, but then we just have one component.

05:50 So work out derivatives in del cross f.

06:02 Okay, component.

06:08 Okay, and i am going to leave out the f knot.

06:12 So just a warning, i'll put it back in at the end.

06:17 Let's see.

06:18 So the partial of fy with respect to x is kind of a mess.

06:30 Yeah, lots of little parts to it.

06:32 It's y squared over a cubed plus y squared over a cubed times e to the xy over a squared over a squared, that's a mess, plus xy over a squared, plus xy over a squared, plus one over a, e to the xy over a squared, plus one over a, e to the xy over a squared.

07:16 Okay, so let me just check that out since there are lots of parts to this.

07:25 So let's see...

Please give Ace some feedback