Chapter 12: Q 63. (page 965)

Prove that $‖ \nabla f (x, y, z) ‖ = 1$ for every point in the domain of the function $f (x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}$ except the origin

Short Answer

Expert verified

Solving $\nabla f = i \frac{\partial f}{\partial x} + j \frac{\partial f}{\partial y} + k \frac{\partial f}{\partial z}$ will prove the result.

Step by step solution

Given Information

It is given that

$f (x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}$

Taking the divergence

Solving LHS

$‖ \nabla f (x, y, z) ‖$

$\nabla f = i \frac{\partial f}{\partial x} + j \frac{\partial f}{\partial y} + k \frac{\partial f}{\partial z}$

$\frac{\partial f}{\partial x} = \frac{2 x}{2 \sqrt{x^{2} + y^{2} + z^{2}}}$

$\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}$

and

$\frac{\partial f}{\partial y} = \frac{2 y}{2 \sqrt{x^{2} + y^{2} + z^{2}}}$

$\frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}$

also

$\frac{\partial f}{\partial z} = \frac{2 z}{2 \sqrt{x^{2} + y^{2} + z^{2}}}$

$\frac{\partial f}{\partial z} = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$

Simplification

Solving, we get

$‖ \nabla f (x, y, z) ‖ = ||i \frac{\partial f}{\partial x} + j \frac{\partial f}{\partial y} + k \frac{\partial f}{\partial z}||$

$‖ \nabla f (x, y, z) ‖ = ||i (\frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}) + j (\frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}) + k (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})||$

$‖ \nabla f (x, y, z) ‖ = \sqrt{{(\frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}})}^{2} + {(\frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}})}^{2} + {(\frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}})}^{2}}$

$‖ \nabla f (x, y, z) ‖ = \sqrt{\frac{x^{2}}{x^{2} + y^{2} + z^{2}} + \frac{y^{2}}{x^{2} + y^{2} + z^{2}} + \frac{z^{2}}{x^{2} + y^{2} + z^{2}}}$

$‖ \nabla f (x, y, z) ‖ = \sqrt{\frac{x^{2} + y^{2} + z^{2}}{x^{2} + y^{2} + z^{2}}}$

$‖ \nabla f (x, y, z) ‖ = 1$

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Prove that $‖ \nabla f (x, y, z) ‖ = 1$ for every point in the domain of the function $f (x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}$ except the origin

Short Answer

Step by step solution

Given Information

Taking the divergence

Simplification

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Prove that ‖∇f(x,y,z)‖=1for every point in the domain of the functionf(x,y,z)=x2+y2+z2except the origin

Short Answer

Step by step solution

Given Information

Taking the divergence

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Prove that $‖ \nabla f (x, y, z) ‖ = 1$ for every point in the domain of the function $f (x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}$ except the origin