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.
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.
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.
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.
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.