Determine if the given vector field F is conservative or...

Determine if the given vector field F is conservative or not. F = ?-3y, 15y^2 - 3z^2 - 3x - 3z, -6yz - 3y? conservative not conservative If F is conservative, find all potential functions f for F so that F = ?f. (If F is not conservative, enter NOT CONSERVATIVE. Use C as an arbitrary constant.) f(x, y, z) =

Transcript

00:01 Hi, let's start the solution.

00:02 In this question we have given vector field f that is equal to minus 3y comma 15y square minus 3z square minus 3x minus 3z comma minus 6yz minus 3y.

00:24 Then curl f equal to vector i j k del upon del x del upon del y del upon del z.

00:38 This one is minus 3y 15y square minus 3z square minus 3x minus 3z and this one is minus 6yz minus 3y.

00:51 Now by definition of curl f that is equal to vector i del upon del y of minus 6yz minus 3y minus del upon del z of 15y square minus 3z minus 3x minus 3z minus vector j del upon del x of minus 6yz minus 3y minus del by del z of minus 3y plus vector k del upon del x of 15y square minus 3z square minus 3x minus 3z minus del upon del y of minus 3y.

01:44 So we get that is equal to vector i minus 6z minus 3 minus minus 6z minus 3 minus vector j 0 minus 0 plus vector k minus 3 minus minus 3.

02:02 So we get that is equal to 0 vector i plus 0 vector j plus 0 vector k that is equal to 0.

02:14 So here we get curl of vector field f is 0 therefore vector field f is conservative.

02:32 Now find a now to find a potential function for f such that f equal to delta f delta f that is we have f is minus 3y vector i plus 15y square minus 3z square minus 3x minus 3z into vector j plus minus 6yz minus 3y into vector k that is equal to del f upon del x vector i plus del f upon del y vector j plus del f upon del z of vector k.

03:26 Now del f upon del x is minus 3y del f upon del y is 15y square minus 3x square minus 3x minus 3z and del f upon del z equal to minus 6yz minus 3y...

Question

Please give Ace some feedback