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) = (-16x+2y)i + (2x+10y)j
f(x, y) =
b. f(x, y)= -8yi - 7xj
f(x, y) =
c. f(x, y) = (-8 sin y)i + (4y - 8x cos y)j
f(x, y) =

A. F is conservative, and B. NC. F is conservative, and

Step-by-step explanation:

Remember that a vector field

is conservative if .

A. In this case, and  . The partial derivatives are and  . Note that therefore F is conservative. By definition, a potential function f must satisfy and . Integrating respect to x on the first equation, where c(y) is a function constant on x. To find c(y), we differentiate respect to y and use the previous equation to obtain . Then so for some constant k. Now . We need that f(0,0)=0 so then k=0 and  B. Here,   and  . Because the partial derivatives and   are not equal, F is not conservative.C. We have that and  , so the partial derivatives are and  . The partial derivartives are equal, therefore F is conservative. Using the same method from part A, a potential function f satisfies and . Integrate respect to x on the first equation to get for a function b(y) constant in x. Now differentiate respect to y to obtain . Then so for a constant t. Because f(0,0)=0, , then t=0 and .