Question

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, ˆ‡f = f) with f(0,0) = 0. If it is not conservative, type n.

a. f(x, y) = (-8x - 7y)i + (-7x + 8y)j
b. f(x, y) = -4yi - 3xj
c. f(x, y) = (-4sin(y))i + (-14y - 4xcos(y))j

Answer 1

To determine whether a vector field is conservative or not, we compute the appropriate first-order partial derivatives. If the derivatives are equal, the vector field is conservative. Using this method, we find that vector fields (a) and (c) are conservative, while vector field (b) is not.

Step-by-step explanation:

To determine whether a vector field is conservative or not, we need to compute the appropriate first-order partial derivatives. If the derivative of the y-component of the force with respect to x is equal to the derivative of the x-component of the force with respect to y, then the vector field is conservative.

Let's calculate the derivatives for each vector field:

1. f(x, y) = (-8x - 7y)i + (-7x + 8y)j:The derivatives are equal, so the vector field is conservative.
   Potential function: f(x, y) = -4x^2 - 7xy + 4y^2
2. f(x, y) = -4yi - 3xj:The derivatives are both zero, so the vector field is conservative.
   Potential function: f(x, y) = 0
3. f(x, y) = (-4sin(y))i + (-14y - 4xcos(y))j:The derivatives are equal, so the vector field is conservative.
   Potential function: f(x, y) = -4ycos(y) + 2xsin(y)