Q1.11P - Introduction to Electrodynamics

Question

Find the gradients of the following functions:

(a)  $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{y}}^{\mathbf{3}}\mathbf{+}{\mathit{z}}^{\mathbf{4}}$

(b)  $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}{\mathit{z}}^{\mathbf{4}}$

(c) $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\left(y\right)\mathit{l}\mathit{n}\left(z\right)$ 

Step-by-Step Solution

Answer

(a) The gradient of the function is $\left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}$.

(b) The gradient of the function is $\left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}$ .

(c) The gradient of the function is $\left(e^x\sin y\ln z\right)\hat{x}+\left(e^x\cos y\ln z\right)\hat{y}+\left(\frac{e^x\sin y}{z}\right)\hat{z}$.

Step 1: Write the given information.

 

The given functions are,

(a)  $f\left(x,y,z\right)=x^4+y^3+z^4$

(b)  $f\left(x,y,z\right)=x^2{y}^3{z}^4$

(c)  $f\left(x,y,z\right)=e^x\sin \left(y\right)\ln \left(z\right)$

Step 2: Define gradient of the function.

The gradient of the function is defined as its slope on the curve of that function for the given particular point.

 

Step 3: Solve for the gradient of part (a).

Write the given function.

 $f\left(x,y,z\right)=x^4+y^3+z^4$

Differentiate the above function as,

 $$\begin{gathered} \frac{\partial f}{\partial x}=2x \\ \frac{\partial f}{\partial y}=3y^2 \\ \frac{\partial f}{\partial z}=4z^3 \end{gathered}$$

Then, the gradient of the function is written as,

 $\begin{array}{rcl}\nabla f& =& \frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}\hat{y}+\frac{\partial f}{\partial z}\hat{z}\\ & =& \left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}\\ & & \end{array}$

Therefore, the gradient of the function is $\left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}$ .

Step 4: Solve for the gradient of part (b).

Write the given function.

 $f\left(x,y,z\right)=x^2{y}^3{z}^4$

Differentiate the above function as,

 $$\begin{gathered} \frac{\partial f}{\partial x}=2x{y}^2{z}^4 \\ \frac{\partial f}{\partial y}=3x^2{y}^2{z}^4 \\ \frac{\partial f}{\partial z}=4x^2{y}^3{z}^3 \end{gathered}$$

Then, the gradient of the function is written as,

 $\begin{array}{rcl}\nabla f& =& \frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}\hat{y}+\frac{\partial f}{\partial z}\hat{z}\\ & =& \left(2x{y}^2{z}^4\right)\hat{x}+\left(3x^2{y}^2{z}^4\right)\hat{y}+\left(4x^2{y}^3{z}^3\right)\hat{z}\\ & & \end{array}$

Therefore, the gradient of the function is $\left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}$.

Step 5: Solve for the gradient of part (c).

Write the given function.

 $f\left(x,y,z\right)=e^x\sin \left(y\right)\ln \left(z\right)$ 

Differentiate the above function as,

 $$\begin{gathered} \frac{\partial f}{\partial x}=e^x\sin y\ln z \\ \frac{\partial f}{\partial y}=e^x\cos y\ln z \\ \frac{\partial f}{\partial z}=\frac{e^x\sin y}{z} \end{gathered}$$

Then, the gradient of the function is written as,

 $\begin{array}{rcl}\nabla f& =& \frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}\hat{y}+\frac{\partial f}{\partial z}\hat{z}\\ & =& \left(e^x\sin y\ln z\right)\hat{x}+\left(e^x\cos y\ln z\right)\hat{y}+\left(\frac{e^x\sin y}{z}\right)\hat{z}\\ & & \end{array}$ 

Therefore, the gradient of the function is .

 $\left(e^x\sin y\ln z\right)\hat{x}+\left(e^x\cos y\ln z\right)\hat{y}+\left(\frac{e^x\sin y}{z}\right)\hat{z}$