Describe the gradient of a function of 3 variables

[kosovo dave]

Homework Statement

Match the function with the description of its gradient.

Homework Equations

f(x,y,z)=√(x^2+y^2+z^2)
1. constant, parallel to xy plane
2. constant, parallel to xz plane
3. constant, parallel to yz plane
4. radial, increasing in magnitude away from the origin
5. radial, constant magnitude
6. radial, decreasing in magnitude away from origin

The Attempt at a Solution

grad f(x,y,z)=(df/dx)i+(df/dy)j+(df/dz)k
grad f=[(x^2+y^2+z^2)^-.5](xi+yj+zk)

I know it's definitely radial. I found a solution online that said the magnitude was constant though, and I can't tell why.

 

[Dick]

Homework Statement

Match the function with the description of its gradient.

Homework Equations

f(x,y,z)=√(x^2+y^2+z^2)
1. constant, parallel to xy plane
2. constant, parallel to xz plane
3. constant, parallel to yz plane
4. radial, increasing in magnitude away from the origin
5. radial, constant magnitude
6. radial, decreasing in magnitude away from origin

The Attempt at a Solution

grad f(x,y,z)=(df/dx)i+(df/dy)j+(df/dz)k
grad f=[(x^2+y^2+z^2)^-.5](xi+yj+zk)

I know it's definitely radial. I found a solution online that said the magnitude was constant though, and I can't tell why.

Well, what is the magnitude of the grad f you computed? What the magnitude of (xi+yj+zk)?

 

[kosovo dave]

Oh I think I get it now. I'd end up with sqrt(x^2+y^2+z^2)/sqrt(x^2+y^2+z^2) just leaving the vector i+j+k?

 

[Dick]

Oh I think I get it now. I'd end up with sqrt(x^2+y^2+z^2)/sqrt(x^2+y^2+z^2) just leaving the vector i+j+k?

Almost, you want to find |grad f|. You replaced the vector with its magnitude. You are left with just 1.

 

[kosovo dave]

Clear as the Mississippi! Just kidding. I get it now. Thanks for the help, Dick!