Question

Decide if the given vector field is the gradient of a function f. If so find f. (Remember to use absolute values where appropriate. If an answer does not exist, enter DNE.) i/x + j/y + k/z f(x, y, z) = If not, explain why not.

A. i/x + j/y + k/z is the gradient of a function.
B. i/x + j/y + k/z is not the gradient of a function because the curl of the field is not equal to zero.
C. 1/x + j/y + k/z is not the gradient of a function because the field is not path independent.
D. i/x + j/y + k/z is not the gradient of a function because the field has no potential function
E. i/x + j/y + k/z is not the gradient of a function because integral_C F middot dr notequalto 0 for every closed curve C.

Answer 1

The given vector field is the gradient of a function. The function is f(x, y, z) = ln| x | + ln| y | + ln| z | + C.

Step-by-step explanation:

The given vector field is i/x + j/y + k/z. To determine if it is the gradient of a function, we need to check if the curl of the field is equal to zero.

The curl of a vector field is given by the formula: curl(F) = (∂Q/∂y - ∂P/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂R/∂x - ∂Q/∂y)k.

In this case, P = i/x, Q = j/y, and R = k/z. Taking the partial derivatives, we have: ∂P/∂y = 0, ∂Q/∂z = 0, ∂R/∂x = 0.

Therefore, the curl of the field is zero in all three components, which means it is the gradient of a function. The function is given by integrating the components with respect to their variables: f(x, y, z) = ∫ i/x dx + ∫ j/y dy + ∫ k/z dz.

Using the antiderivative, we find that f(x, y, z) = ln| x | + ln| y | + ln| z | + C, where C is the constant of integration.