Extending the Derivative

We have seen how to differentiate functions of one variable, but what about functions of multiple variables? We've already seen that functions can have multiple inputs and multiple outputs, and they appear frequently in the real world. In this section, we will extend the concept of the derivative to functions of multiple variables.

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📄️ The Partial Derivative

The Partial Derivative is a way to think about the derivative of a function of multiple variables.

📄️ Second Partial Derivatives

Just like how we can take second derivatives of functions of one variable, we can also take second derivatives of functions of two variables.

📄️ Gradient

The gradient of a function has multiple interpretations and uses in mathematics.

📄️ The Directional Derivative

The directional derivative is somewhat of an extension or generalization of the partial derivative.

📄️ Differentiating Vector-Valued Functions

So far, we've been dealing with functions that output a single value.

📄️ Multivariable Chain Rule

The chain rule is a fundamental concept in calculus, and it can be extended to multivariable functions.

📄️ Gradient of a Function in Cylindrical Coordinates

The gradient in cylindrical coordinates is defined a bit differently than in Cartesian coordinates.

📄️ Partial Derivatives in Cylindrical Coordinate Systems

Previously, we discussed the partial derivative of a function in Cartesian coordinates.

Overview

First, we discuss the partial derivative, which is the derivative of a function with respect to one of its variables, holding the other variables constant. We extend some of our rules from single-variable calculus to partial derivatives, one of which is the chain rule. We introduce second partial derivatives, and touch on the symmetry of mixed partial derivatives.

Next, we discuss the gradient of a function, which from the surface looks like a vector of partial derivatives. However, through the concept of the directional derivative, we gain deeper insight into the physical meaning of the gradient.

After that, we take a detour into a concept called curvature, and fully go through its lengthy derivation. On the way, we learn important concepts like the tangent vector and arc lengths, which are crucial for other topics in multivariable calculus.

With the groundwork laid, we move on to some higher-level constructs. We introduce the divergence, curl, and Laplacian operators, and take a look at their applications through understanding Maxwell's equations.

Finally, we discuss the concept of the Jacobian, which is a generalization of the derivative to functions of multiple variables.