Question

For each of the following, decide if the given vector field is a gradient of a function. If so, find the function and enter it as your answer. If not, enter the word none for your answer and be sure that you are able to explain why this is the case.

Answer 1

To determine if a vector field is the gradient of a function, one must check if it is conservative by verifying if the mixed partial derivatives of its components are equal. If they are equal, the field is conservative and can be integrated to find the potential function; otherwise, the answer is none.

Step-by-step explanation:

In general, a vector field is a gradient if it satisfies the following conditions:

Conservative Field: The vector field must be conservative. This means that the line integral of the vector field around any closed loop is zero.

Mathematically, if F is the vector field, then for any closed curve C, the line integral ∮CF⋅dr=0.

Existence of Potential Function: If a vector field is conservative, it can be expressed as the gradient of a scalar function. If F=∇f, where f is a scalar function, then F is a gradient field.

If you have specific vector fields you'd like me to examine, please provide them, and I can give you a detailed explanation for each.