## 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.