Question

decide whether or not the given vector field could be a gradient vector field. Give a justification for your answer. F(x,y)= I am at the point in my textbook that I am not yet supposed to use the line curl test. Can anyone please help because I feel that I'm missing something.

Answer 1

You could integrate the functions that define the x and y components of the field and see if they are the x and y derivatives of the same fct......

Answer 2

first of all thank you for being the first to answer my question but the problem is that this is one of various problems but isn't there a more powerful way of showing that the line integral is the same independent of path chosen b/c despite choosing two different paths the line integrals could be equal for those two paths, just not all of them

Answer 3

You could show that the mixed partials are equal, but thats really the curl test in disguise.

Answer 4

How about this:

1) Integrate the x-component wrt x, being sure to add the integration constant C(y),
which is a fct of y.
2) Diofferentiate the answer to 1) wrt y and compare it to the y-component. If they differ
by only a fct of y, then you can choose a C(y) that will make them the same, so F is the
gradient of some function U(x,y).

OR(this is the mixed partial test)

If F=(M(x,y),N(x,y)) then F is a gradient if dM/dy=dN/dx.

Answer 5

Oh thanks, I had skipped over the page covering the partial derivative test in my textbook. thanks for the help.