determine whether or not the vector field is conservative. if it is conservative, find a function f such…

determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = 7yz²e^xz i + 7ze^xz j + 7xyze^xz k f(x, y, z) =

determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = 7yz²e^xz i + 7ze^xz j + 7xyze^xz k f(x, y, z) =

Answer

Explanation:

Step1: Comprobar si el campo es conservativo

Un campo vectorial $\mathbf{F}(x,y,z)=M\mathbf{i}+N\mathbf{j}+P\mathbf{k}$ es conservativo si $\nabla\times\mathbf{F}=\mathbf{0}$, donde $M = 7yz^{2}e^{xz}$, $N = 7ze^{xz}$ y $P=7xyze^{xz}$. Calculamos el rotor $\nabla\times\mathbf{F}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\7yz^{2}e^{xz}&7ze^{xz}&7xyze^{xz}\end{vmatrix}$ $\frac{\partial P}{\partial y}-\frac{\partial N}{\partial z}=7xze^{xz}-(7e^{xz}+7xze^{xz})=- 7e^{xz}$ $\frac{\partial M}{\partial z}-\frac{\partial P}{\partial x}=14yze^{xz}+7xyz^{2}e^{xz}-(7yze^{xz}+7xyz^{2}e^{xz}) = 7yze^{xz}$ $\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}=7z^{2}e^{xz}-7z^{2}e^{xz}=0$ Como $\nabla\times\mathbf{F}\neq\mathbf{0}$, el campo vectorial no es conservativo.

Answer:

DNE