Question

compute the gradient vector fields of the following functions:
a. $f(x,y)=9x^{2}+4y^{2}$
$\
abla f(x,y)=\square\mathbf{i}+\square\mathbf{j}$
b. $f(x,y)=x^{6}y^{9}$,
$\
abla f(x,y)=\square\mathbf{i}+\square\mathbf{j}$
c. $f(x,y)=9x + 4y$
$\
abla f(x,y)=\square\mathbf{i}+\square\mathbf{j}$
d. $f(x,y,z)=9x + 4y+6z$
$\
abla f(x,y)=\square\mathbf{i}+\square\mathbf{j}+\square\mathbf{k}$
e. $f(x,y,z)=9x^{2}+4y^{2}+6z^{2}$
$\
abla f(x,y,z)=\square\mathbf{i}+\square\mathbf{j}+\square\mathbf{k}$

Explanation:

Step1: Recall the gradient formula

For a function \(f(x,y)\), \(
abla f(x,y)=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}\). For a function \(f(x,y,z)\), \(
abla f(x,y,z)=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}\)

Step2: Solve part A

Given \(f(x,y) = 9x^{2}+4y^{2}\)

• \(\frac{\partial f}{\partial x}=\frac{\partial(9x^{2}+4y^{2})}{\partial x}=18x\)
• \(\frac{\partial f}{\partial y}=\frac{\partial(9x^{2}+4y^{2})}{\partial y}=8y\)

Step3: Solve part B

Given \(f(x,y)=x^{6}y^{9}\)

• \(\frac{\partial f}{\partial x}=\frac{\partial(x^{6}y^{9})}{\partial x}=6x^{5}y^{9}\)
• \(\frac{\partial f}{\partial y}=\frac{\partial(x^{6}y^{9})}{\partial y}=9x^{6}y^{8}\)

Step4: Solve part C

Given \(f(x,y)=9x + 4y\)

• \(\frac{\partial f}{\partial x}=\frac{\partial(9x + 4y)}{\partial x}=9\)
• \(\frac{\partial f}{\partial y}=\frac{\partial(9x + 4y)}{\partial y}=4\)

Step5: Solve part D

Given \(f(x,y,z)=9x + 4y+6z\)

• \(\frac{\partial f}{\partial x}=\frac{\partial(9x + 4y+6z)}{\partial x}=9\)
• \(\frac{\partial f}{\partial y}=\frac{\partial(9x + 4y+6z)}{\partial y}=4\)
• \(\frac{\partial f}{\partial z}=\frac{\partial(9x + 4y+6z)}{\partial z}=6\)

Step6: Solve part E

Given \(f(x,y,z)=9x^{2}+4y^{2}+6z^{2}\)

• \(\frac{\partial f}{\partial x}=\frac{\partial(9x^{2}+4y^{2}+6z^{2})}{\partial x}=18x\)
• \(\frac{\partial f}{\partial y}=\frac{\partial(9x^{2}+4y^{2}+6z^{2})}{\partial y}=8y\)
• \(\frac{\partial f}{\partial z}=\frac{\partial(9x^{2}+4y^{2}+6z^{2})}{\partial z}=12z\)

Answer:

A. \(18x\mathbf{i}+8y\mathbf{j}\)
B. \(6x^{5}y^{9}\mathbf{i}+9x^{6}y^{8}\mathbf{j}\)
C. \(9\mathbf{i}+4\mathbf{j}\)
D. \(9\mathbf{i}+4\mathbf{j}+6\mathbf{k}\)
E. \(18x\mathbf{i}+8y\mathbf{j}+12z\mathbf{k}\)