# Calculus 3

## Gain expertise in Calculus 3 with these articles and videos that explain everything (step-by-step) on calculus in three dimensions.

Calculus 3 is a high-school or undergraduate mathematics course. This course is the third in a series that introduces students to calculus and usually begins with vector functions and functions of several variables. After that, instructors consider limits and continuity of functions. Then students study partial derivatives and multiple integrals. In particular, Lagrange multipliers and integration using a transformation between coordinate systems are favorites. Finally, in the end, students explore the most beautiful theorems in all of calculus, including Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.

## Derivatives and Integrals of Vector Functions (and Tangent Vectors) [Video]

Okay, so after learning what vector functions are by sketching some graphs and using vector functions operations, let’s do some calculus with vector functions. In this video, I explain how to take derivatives and integrals of vector functions. I also discuss tangent vectors in detail.

## Vector Functions and Space Curves (Calculus in 3D) [Video]

Vector functions of real variables are defined and studied. I begin with an unintuitive introduction to vector functions and how they can model motion in space. I discuss the graph of a vector function and how to determine a vector function given geometric information. Operations with vector functions, limits of vector functions, and continuity of vector functions, are also what I explore.

## Green’s Theorem (by Example)

In this article, I discuss Green’s theorem for simply connected regions and doubly connected regions. I provide many examples for both cases.

## Probability Density Functions (Applications of Integrals)

In this article, I will consider the following applications of multiple integrals: the average value of a function over a region, the mass of a lamina, electric charge, moments and center of mass, moments of inertia, and probability density functions.

## Conservative Vector Fields and Independence of Path

The concept of conservative vector fields allows us to generalize the fundamental theorem of calculus to line integrals. I discuss the Fundamental Theorem of Line Integrals, work in a conservative vector field, and then finding an area using a line integral.

## Line Integrals (Theory and Examples)

In this article, I go over what a line integral is, and then I cover evaluating line integrals using parametrization. After that, I discuss line integrals with respect to coordinate variables and the line integral of a vector field along a curve. Find out more about line integrals.

## Divergence and Curl of a Vector Field

Okay, so now you know what a vector field is, what operations can you do on them? In this article, I explain the many properties of the divergence and the curl and work through examples.

## Vector Fields and Gradient Fields

You’ve heard of an electric field, a magnetic field, or a gravitational field. What is a vector field? You’ve learned about vector functions of a single variable and multivariable functions. Now it’s time to start putting everything together. In this article, I cover vector fields and various types of vector fields.

## Jacobian (Change of Variables in Multiple Integrals)

Okay, so by now, you have seen how to determine integrals using polar coordinates, cylindrical and spherical coordinates. But how do you come up with your coordinate system so that an integral becomes much easier to determine? In this article, I cover the Jacobian and how to make a change of variable in a double integral.

## Triple Integrals in Cylindrical and Spherical Coordinates

Okay, so you know double integrals and how to work with them in polar coordinates; now, it’s time to learn triple integrals in cylindrical and spherical coordinates. You will find many examples that I detail here.

## Triple Integrals (With Fubini’s Theorem)

Can you perform integration with more than one or two integrals? In this article, I discuss triple integrals and their basic properties. After explaining different types of integration regions, I cover Fubini’s theorem for triple integrals and some applications.

## Surface Area Using Double Integrals

How do you find the surface area of a torus? In this article, I discuss finding surface area of differentiable functions. After working out several examples, I then explain how to find the surface area that is defined parametrically, such as a torus.

## Double Integrals in Polar Coordinates

Okay, so you know what a double integral is and how to use Fubini’s theorem. In this article, I show you how to use Fubini’s theorem in polar coordinates. I work through several examples and provide step-by-step reasoning.

## Fubini’s Theorem for Double Integrals

Did you notice how working with a double integral as a limit of a Riemann sum is very tedious? This article covers iterated integrals and how to find them using a celebrated theorem: Fubini’s Theorem. I discuss precisely when this theorem applies and how-to set up an integral by considering vertically simple and horizontal simple regions.

## Double Integrals and the Volume Under a Surface

In this article, I motivate the double integral using partitions and Riemann sums. Then, I cover the definition of the double integral and provide examples. After that, I cover the basic properties of the double integral.

## Lagrange Multipliers (Optimizing a Function)

Do you know who Lagrange was? Do you know why his theorem is so famous? In this article, I go over the Lagrange Theorem, which is used in a procedure (also named after Lagrange) called Lagrange Multipliers. Find out what all this means.

## Absolute Extrema (and the Extreme Value Theorem)

Finding the extrema of a function is the quintessential calculus problem. But exactly how do we approach this topic with multivariable functions? Fund out here, and learn the celebrated Extreme Value Theorem. I also cover several compelling examples with interesting boundary conditions.

## Relative Extrema and the Second Partials Test

Okay, so you know the First and Second derivative test for finding relative extrema. How can we use partial derivatives to find extrema of functions of two or more variables? In this article, I motivate critical points, saddle points, and the need for a derivative test for finding extrema. You will also find the Second Partials Test with examples.

## Normal Lines and Tangent Planes

If a multivariable function is differentiable, then the tangent plane exists. But how do you find an equation for it? I explain how to do this, and I discuss normal lines as well. You will see that I worked through several examples.

## Directional Derivatives and Gradient Vectors

The directional derivative generalizes the partial derivative. In this article, I motivate the directional derivative and discuss the gradient of a function. Towards the end, I cover the steepest ascent and steepest descent and provide examples.