Expert Answer to "Find the gradient vector field of f.f(x,y,z)=(x2+y2+z2)12"

Zoe Oneal

Answered question

2021-10-14

Find the gradient vector field of f.

f (x, y, z) = {(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}

Answer & Explanation

Yusuf Keller

Skilled2021-10-15Added 90 answers

Step 1
We want to find the gradient field of the function
$f (x, y, z) = \sqrt{(x^{2} + y^{2} + z^{2})}$
Recall that,
$▽ f = \frac{\partial f}{\partial x} (x, y, z) i + \frac{\partial f}{\partial y} (x, y, z) j + \frac{\partial f}{\partial z} (x, y, z) k$
$= \frac{\frac{\partial (x^{2} + y^{2} + z^{2})}{\partial x}}{\sqrt{x^{2} + y^{2} + z^{2}}} i + \frac{2 \frac{\partial (x^{2} + y^{2} + z^{2})}{\partial y}}{2 \sqrt{x^{2} + y^{2} + z^{2}}} j + \frac{\frac{\partial (x^{2} + y^{2} + z^{2})}{\partial z}}{2 \sqrt{x^{2} + y^{2} + z^{2}}} k$
$= \frac{2 x}{2 \sqrt{x^{2} + y^{2} + z^{2}}} i + \frac{2 y}{2 \sqrt{x^{2} + y^{2} + z^{2}}} j + \frac{2 z}{2 \sqrt{x^{2} + y^{2} + z^{2}}} k$
$= \frac{x}{x^{2} + y^{2} + z^{2}} i + \frac{y}{x^{2} + y^{2} + z^{2}} + \frac{z}{x^{2} + y^{2} + z^{2}}$
Step 2
Hence the gradient vector field of f is
$▽ f = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} i + \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} + \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$
Result
$▽ f = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} i + \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} + \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$

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