Griffiths 1.11 Find the gradients of the following functions:

Griffiths 1.11 Find the gradients of the following functions:


P
Asked by 2 years ago
945 points

Find the gradients of the following functions:

(a) $f(x,y,z)=x^2+y^3+z^4$

(b) $f(x,y,z)=x^2y^3z^4$

(c) $f(x,y,z)=e^x \sin{(y)}\ln{(z)}$

Griffiths 1
Pairoaj

1 Answer

P
Answered by 2 years ago
945 points

The definition of the gradient is

$$\nabla f=\frac{\partial f}{\partial x} \hat{x}+\frac{\partial f}{\partial y} \hat{y}+\frac{\partial f}{\partial z} \hat{z}$$

(a)

$$\nabla f=\frac{\partial}{\partial x}(x^2+y^3+z^4) \hat{x}+\frac{\partial }{\partial y}(x^2+y^3+z^4) \hat{y}+\frac{\partial}{\partial z}(x^2+y^3+z^4) \hat{z}$$

$$ =2x \hat{x}+3y^2 \hat{y}+4z^3 \hat{z}$$

proceed in this manner for parts b and c

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Asked: 2 years ago

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