Grad Vector Direction: Clarified with Grad F Surfaces

In summary, grad F (F surface) is in normal direction and (grad F(r)) x r = F'(r) (r) x r = 0 implies grad F is in the direction of r, which is the radial direction. However, the radial and normal directions may not always be the same. When a level surface is defined by F(r)=some constant, where "r" is the radial variable, the level surface will always be a circle and the radial direction will always be parallel to the vector normal to the level surface. The direction of the grad vector can be clarified by understanding this relationship.
  • #1
seshikanth
20
0
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?
Thanks,
 
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  • #2
Can u please throw some light on this?
Thanks
 
  • #3
For a level surface defined by F(r)=some constant, where "r" is the radial variable, then by necessity, those level surface(s) will be circles (in 2-D).
And therefore, the radial direction will always be parallell to the vector normal to the level surface.
 

Related to Grad Vector Direction: Clarified with Grad F Surfaces

1. What is a gradient vector?

A gradient vector is a mathematical concept used in vector calculus and multivariable calculus. It represents the direction and magnitude of the steepest slope of a function at a specific point.

2. How is the direction of a gradient vector determined?

The direction of a gradient vector is determined by the direction of the partial derivatives of the function at a specific point. The gradient vector will point in the direction of the greatest increase in the function.

3. What is the significance of the direction of the gradient vector?

The direction of the gradient vector is significant because it shows the direction in which the function is changing the fastest at a specific point. This information is useful in optimization problems and in understanding the behavior of a function.

4. What are "Grad F surfaces"?

"Grad F surfaces" refer to the surfaces created by plotting the gradient vector of a function at different points. These surfaces can help visualize the direction and magnitude of the gradient vector at different points in the function's domain.

5. How can "Grad F surfaces" be used to understand the behavior of a function?

By looking at the "Grad F surfaces" of a function, one can see the direction of the gradient vector at different points and identify areas of steepest increase or decrease. This information can provide insight into the behavior of the function and aid in optimization and problem-solving.

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