When gradient is parallel to position vector

[oahsen]

Homework Statement

suppose that grad of f(x,y,z) is always parallel to the position vector xi+yj+zk. show that f(0,0,a)=f(0,0,-a) for any a.

The Attempt at a Solution

grad of f= fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k ; then gradf (dot) pos.vector = |gradf|*|pos.vector| (since cos(teta)=1 ) ===> however ı could not do anything with this equation and ı have not any other idea...

 

[Mathgician]

dot product, or cosine of theta

positive if theta is 0, negative if theta is 180 or pi

that tells your max and min slope

 

[oahsen]

dot product, or cosine of theta

positive if theta is 0, negative if theta is 180 or pi

that tells your max and min slope

what should ı do with the max and min slope value?

 

[Mathgician]

what should ı do with the max and min slope value?

what is the vector from f(0,0,a) to f(0,0,-a)?

find the unit vector of this vector,

and then do the dot product of the unit vector and the gradient vector.

 

[oahsen]

what is the vector from f(0,0,a) to f(0,0,-a)?

find the unit vector of this vector,

and then do the dot product of the unit vector and the gradient vector.

since we do not know f how can we find the vector from f(0,0,a) to f(0,0,-a)?
please be more clear. I do not know perfectly this subject.

 

[Mathgician]

since we do not know f how can we find the vector from f(0,0,a) to f(0,0,-a)?
please be more clear. I do not know perfectly this subject.

lets call f(0,0,a) point P, and f(0,0,-a) point S

What is vector PS?

 

[oahsen]

lets call f(0,0,a) point P, and f(0,0,-a) point S

What is vector PS?

is it -2ak (ı am not sure)

 

[Mathgician]

find the unit vector of that and do the dot product of that and the gradiant and see if you get the answer you are looking for