find the gradient of xy^2z^2
Answers
Answer:
Gradient of a function is represented as the partial derivatives of the given scalar function using arithmetic operators.
In terms of formula, it is represented as:
\boxed{ \bf{\nabla \:F = \dfrac{\partial f}{\partial x} + \dfrac{\partial f}{\partial y} + \dfrac{\partial f}{\partial z} }}
∇F=
∂x
∂f
+
∂y
∂f
+
∂z
∂f
Now it is given that, scalar function f(x,y,z) is equal to x² + y³ + z⁴.
Using the formula given above we get:
\begin{gathered}\implies \dfrac{ \partial f}{ \partial x} = \dfrac{ \partial (x^2 + y^3 + z^4)}{ \partial x}\\\\\\\implies \dfrac{ \partial f}{ \partial x} = 2x + 0 + 0 = 2x\end{gathered}
⟹
∂x
∂f
=
∂x
∂(x
2
+y
3
+z
4
)
⟹
∂x
∂f
=2x+0+0=2x
\begin{gathered}\implies \dfrac{ \partial f}{ \partial y} = \dfrac{ \partial (x^2 + y^3 + z^4)}{ \partial y}\\\\\\\implies \dfrac{ \partial f}{ \partial y} = 0 + 3y^2 + 0 = 3y^2\end{gathered}
⟹
∂y
∂f
=
∂y
∂(x
2
+y
3
+z
4
)
⟹
∂y
∂f
=0+3y
2
+0=3y
2
\begin{gathered}\implies \dfrac{ \partial f}{ \partial z} = \dfrac{ \partial (x^2 + y^3 + z^4)}{ \partial z}\\\\\\\implies \dfrac{ \partial f}{ \partial z} = 0 + 0 + 4z^3 = 4z^3\end{gathered}
⟹
∂z
∂f
=
∂z
∂(x
2
+y
3
+z
4
)
⟹
∂z
∂f
=0+0+4z
3
=4z
3
Combining all the three in the formula we get:
\implies \boxed{ \bf{\nabla\:F = 2x + 3y^2 + 4z^3}}⟹
∇F=2x+3y
2
+4z
3
This is the required answer.