Is the Gradient Vector Always in the Radial Direction?

[seshikanth]

As we know grad F (F surface) is in normal direction. But we also have (grad F(r)) x r = F'(r) (r) x r = 0
this implies grad F is in direction of r i.e., radial direction. Radial and normal directions need not be same. Can any öne clarify THE DIRECTION OF GRAD VECTOR?

 

[seshikanth]

Gentle reminder

 

[HallsofIvy]

In your second formula, you refer to "F(r)" so you are assuming that F depends only on r and so is spherically symmetric. The first formula does not assume that .

 

[seshikanth]

Got it! Thanks!

 

[blue_raver22]

The direction of the gradient vector is dependent on the direction of the variable in question. In the case of a scalar field, the gradient vector points in the direction of the steepest increase of the function. This can be thought of as the direction in which the function is changing the most rapidly.

In the example given, the gradient vector is in the normal direction because the function is a surface, and the normal direction is perpendicular to the surface. However, when considering the gradient vector in terms of the variable r, it is in the radial direction because r is a distance from the origin and the gradient vector is pointing towards the direction of greatest increase in the function from that point.

Therefore, the direction of the gradient vector can vary depending on the context and should be interpreted accordingly. It is important to consider the variable in question and the overall context when determining the direction of the gradient vector.