2. Let C be the triangle with vertices (0,0), (3,0), and...

2. Let C be the triangle with vertices (0,0), (3,0), and (0,4), oriented counter-clockwise. Find the (outward) flux of \(\vec{F}(x, y) = \begin{pmatrix} x \\ 2y \end{pmatrix}\) across C by evaluating three line integrals. (c) Flux across the side with vertices (0,0) and (0,4):

Transcript

00:01 In this question, we have been given a vector function which is defined as x comma y.

00:07 And we have been given a triangle c with its vertices at 0 comma plus minus 4 and 5 .0.

00:18 And it is oriented counterclockwise.

00:22 We need to find what is the close integral over this curve f dot dr by parameterizing.

00:31 C.

00:31 Okay, so let us see how can we solve this.

00:33 First of all, let's draw the diagram to understand this.

00:37 This is our coordinate 5 comma 0 and these are the coordinate 0 .4 and here i'm having 0.

00:46 Minus 4.

00:47 So this will be the triangle.

00:50 Orientation is given to be counterclockwise.

00:53 So this is our counterclockwise direction.

00:56 Correct.

00:57 And let me name it v, c, a.

01:00 These are the 3 .5.

01:01 These are the vertices.

01:03 So what i will do i will parameterize each of the lines.

01:06 Okay.

01:07 So parameterize first i'm going to parameterize the line.

01:14 Let's say ca.

01:17 Okay.

01:18 So what about this line if i try to parameterize it? y minus okay.

01:26 So i will consider this as x1 comma y1.

01:31 So y minus y 1 which is dip divided by x minus x1 equals to y2 minus y1 so this i will consider as x2 comma y2 so x2 comma y2 so y2 so y 2 minus y 1 will be what it will be 0 minus 4 divided by 5 minus 0 and i will consider it to be equal to t since we are parameterizing it where the value of t lies between 0 and 1 so let us simplify so y minus 4 divided by x equals to minus 4 divided by 5 which is equal to t...

Question

2. Let C be the triangle with vertices (0,0), (3,0), and (0,4), oriented counter-clockwise. Find the (outward) flux of \(\vec{F}(x, y) = \begin{pmatrix} x \\ 2y \end{pmatrix}\) across C by evaluating three line integrals. (c) Flux across the side with vertices (0,0) and (0,4):

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