What is the relationship between the gradient and level sets of a function?

[pamparana]

Hello,

I just want to confirm with the experts here that I have understood the concept of the gradient correctly.

So, a gradient for a function is a vector field that has the partial derivatives of the function. So, for each point in the domain of the function there is a vector associated and each component of that vector tells us how that function is changing at that point w.r.t to the given variables.

So, if I take a function in 3D space which is parameterized over x, y and z directions then the vector woould have 3 components and each component is telling us how the function is changing in that given direction.

Is this explanation sensible?

Thanks,

Luc

 

[slider142]

Easier to visualize, you may note that the gradient vector of a function is always normal to level sets of the function (ie., level curves, level surfaces, etc.). So if you have a good idea of what the level sets look like, you also have a good idea of the gradient vector field. This can be justified by noting that the directional directive along a tangent vector to a level set should be 0, as the function is constant along the level set.
This view also allows you to see the gradient vectors independently of the coordinate system, which is useful in applications.