Consider the following vector field. f(x, y, z) = 5xyez i yzex k (a) find the curl of the vector field.

Multivariable calculus (also known as multivariate calculus) is the extension of one-variable calculus to multivariable calculus: the differentiation and integration of functions involving multiple variables rather than just one.
Multivariable calculus is a fundamental component of advanced calculus.
Calculus on Euclidean space is an advanced calculus topic.

So,

To find the curve:

[Vector Field] Set up [curl F]:

[tex]Curl F=\left|\begin{array}{ccc}\hat{1} & \hat{\jmath} & \hat{k} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\7 x y e^z & 0 & y z e^x\end{array}\right|[/tex]

Simplify [3x3 Matrix Determinant] with [curl F]:

[tex]Curl F=\left|\begin{array}{cc}\frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\0 & y z e^x\end{array}\right| \hat{i}-\left|\begin{array}{cc}\frac{\partial}{\partial x} & \frac{\partial}{\partial z} \\7 x y e^z & y z e^x\end{array}\right| \hat{j}+\left|\begin{array}{cc}\frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\7 x y e^z & 0\end{array}\right| \hat{k}[/tex]

Simplify [2x2 Matrix Determinant] with [curl F]:

[tex]Curl F=\left[\frac{\partial}{\partial y} y z e^x-\frac{\partial}{\partial z} 0\right] \hat{\mathrm{i}}-\left[\frac{\partial}{\partial x} y z e^x-\frac{\partial}{\partial z} 7 x y e^z\right] \hat{\mathrm{\jmath}}+\left[\frac{\partial}{\partial x} 0-\frac{\partial}{\partial y} 7 x y e^z\right] \hat{\mathrm{k}}[/tex]

The partial derivatives can be differentiated using the basic differentiation techniques listed above under "Calculus":

[tex]\begin{aligned}&\frac{\partial}{\partial y} y z e^x=z e^x \\&\frac{\partial}{\partial z} 0=0 \\&\frac{\partial}{\partial x} y z e^x=y z e^x \\&\frac{\partial}{\partial z} 7 x y e^z=7 x y e^z \\&\frac{\partial}{\partial x} 0=0 \\&\frac{\partial}{\partial y} 7 x y e^z=7 x e^z\end{aligned}[/tex]

When we substitute our partial derivative values, we get:

[tex]\begin{aligned}Curl F&=\left(z e^x-0\right) \hat{1}-\left(y z e^x-7 x y e^z\right) \hat{\jmath}+\left(0-7 x e^z\right) \hat{k} \\&=z e^x \hat{\imath}-\left(y z e^x-7 x y e^z\right) \hat{\jmath}-7 x e^z \hat{k}\end{aligned}[/tex]

So, we've discovered the curl of the given vector field.

Therefore, the curl of the vector field is:

[tex]=z e^x \hat{\imath}-\left(y z e^x-7 x y e^z\right) \hat{\jmath}-7 x e^z \hat{k}[/tex]

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The correct question is given below:
Consider the following vector field. f(x, y, z) = 7xyezi + yzexk (a) find the curl of the vector field.