(1 point) For each of the following vector fields F , decide...

(1 point) For each of the following vector fields F , decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, ?f = F) with f(0, 0) = 0. If it is not conservative, type N. A. F (x, y) = (?6x ? 7y) i + (?7x + 14y) j f (x, y) = B. F (x, y) = ?3yi ? 2xj f (x, y) = C. F (x, y) = (?3 sin y) i + (?14y ? 3x cos y) j f (x, y) =

Transcript

00:01 Question we say for each of the following vector fields f we want to determine whether it is conservative or not by computing the appropriate first order partial derivatives.

00:10 Then we're going to type in a potential function little f such that the gradient of little f is equal to capital f with f of 0 0 equaling 0 if it is conservative and if it's not conservative we type n.

00:23 So i want to just review the criteria for being conservative which is down here in blue.

00:30 If we have a vector field f which is pi plus qj then in order for the vector field to be conservative it must be true that the partial of p with respect to y is equal to the partial of q with respect to x.

00:46 So let's take a look at vector field a.

00:50 So in a we have p and we have q right.

00:54 So first of all what is my partial of p with respect to y? that would be 6.

01:02 How about my partial of q with respect to x? that would be 6 as well.

01:09 So this is conservative.

01:12 That means this is the gradient of some potential function f.

01:18 So f sub x is equal to negative 16 x plus 6 y while f sub y is 6 x plus 16 y.

01:31 So what does this tell you about f? well if i integrate with respect to x i get negative 8 x squared plus i'm gonna have 6 x y.

01:45 Whereas over here and it's plus some other function that is a function of y alone so plus some g over y.

01:55 Whereas over here i'm integrating with respect to y.

01:58 I'm gonna get 6 x y plus 8 y squared plus some function of x.

02:08 And so if i put this together what am i getting for f of x y? i'm getting negative 8 x squared and then it's plus 6 x y and then it's plus 8 y squared.

02:26 That would be my answer for part a.

02:30 Now in b i have f of x y equals negative 8 y minus 7 x...

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