# Geometric interpretation of partial derivatives

• MexChemE
In summary, 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? Partial derivatives are used to calculate the rate of change of a function with respect to one or more variables.
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?

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.

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.

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

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!

## 1. What is the geometric interpretation of partial derivatives?

The geometric interpretation of partial derivatives is the rate of change of a function in a specific direction. It represents the slope of the tangent line to the function's graph at a given point in that direction.

## 2. How are partial derivatives represented graphically?

Partial derivatives are usually represented graphically using contour plots. These plots show the level curves of the function, and the direction of steepest ascent or descent at any point is given by the normal vector to the contour curve.

## 3. What is the significance of partial derivatives in multivariate calculus?

Partial derivatives play a crucial role in multivariate calculus as they allow us to analyze how a function changes in multiple variables simultaneously. They are also important in optimization problems, where we want to find the maximum or minimum of a function.

## 4. How do partial derivatives relate to total derivatives?

Partial derivatives are the building blocks of total derivatives. Total derivatives are used to measure the overall change in a function when all of its variables are varied. They are calculated by taking the sum of the partial derivatives in each variable multiplied by the corresponding change in that variable.

## 5. Can partial derivatives be negative?

Yes, partial derivatives can be negative. This means that the function is decreasing in that specific direction. However, the magnitude of the partial derivative (the absolute value) is what tells us how steep the slope is, regardless of its direction.

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