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Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative.$$\quad \mathbf{F}=\langle x, 2\rangle \text { from } A(0,0) \text { to } B(2,4)$$

10

Calculus 3

Chapter 17

Vector Calculus

Section 3

Conservative Vector Fields

Vectors

Vector Functions

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So in this video, we're asked to determine the work. Um, you don't worry, uh, or skilled along the curb or the path Well, 00 to 4. First thing we're gonna do so the work will be the equivalent of determining relating to grow, so it will help us. A lot of our field is conservative. So first of all this, check it for feel this conservative feel the way you do that is you, James. The little left by the line is equal to deliver gee, by DX. So the derivative of X with respect to y is zero and the derivative off to a respective X is also zero. So I feel disconcerted now. Since our field is conservative, we know that it is the ingredient of a potential function. So we know that X Comma two is the radiant of some potential function. And let's try to determine with the potential function. So I need something was derivative with respect to X is X Well, we think about this for a bit, but realize that one her x squared, the derivative of 1/2 x squared That perspective axes just x. All right now I need something was derivative with respect. A Why is too again if we think about this two y pitch that cracked Iria. Why that criteria? Why? Because if you take the derivative of to and respect to, why vested, too. All right, so we determined our potential function, and this helps us a lot. Why? Because this is just equivalent to so to find the work. I just need to determine the line Integral So and since there feel this conservative, it's just it's just the difference between the potential of the points. So he started 00 nly voted to four, so we need to determine fee it to four feet 00 and then subtracted from each other. So if we plug in to for exit of potential function and for for why, in our potential function we get two square divided by two plus two times eight. Sorry, two times four two times floor and then minus here. We're just gonna have zero. So to swear, delighted by two, it's two plus eight, it's 10 so the work is simply equal to 10

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