Geometric interpretation of partial derivatives

[MexChemE]

Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to one variable is the slope of the tangent line to some point of that function.

Also, if a partial derivative is the rate of change of just one variable in a function of several variables, is there a way to calculate the total rate of change of the function?

Thanks in advance!

 

[FactChecker]

I agree with your confusion. You may want to ask him for clarification rather than speculating. He may be talking about functions that are not real valued. Or he may just mean that the concept of slope gets more complicated. A real valued function of several variables can have the partial derivatives that you mention and a gradient. A lot of mathematics related to linear and nonlinear optimization uses the concepts of partial derivatives, gradients, slopes, and direction of steepest descent.

 

[Mark44]

A partial derivative represents the slope of the tangent line to a surface, in one plane. For example, if z = f(x, y), $\frac{\partial f}{\partial x}$ represents the slope of the tangent plane to the surface z = f(x, y), in the x-z plane. The description for $\frac{\partial f}{\partial y}$ is similar.

 

[atyy]

Perhaps he just meant that if we are in Euclidean space and we choose an origin, a point in space is represented by a vector from that origin. However, we don't think of a velocity vector as having the same origin as a point in space. The origin of a velocity vector is attached to some part of a trajectory, and the velocity vector belongs to a tangent space.

When space is curved, points in space are no longer represented as a vector from an origin, and the position vector space is not a meaningful any more. But the tangent spaces are still meaningful concepts.

Here is some information about tangent vectors and partial derivatives: http://mathworld.wolfram.com/ManifoldTangentVector.html

 

[MexChemE]

I did not understand the last post that much, my math is not at that level yet. My doubt has been solved, however. I understand now what a partial derivative represents. Thanks everyone!