A basic tutorial on how to determine the center of mass of a thin plate. The technique involves a double integral of the density function and is a common problem in applied mathematics, physics and engineering.
This is basic tutorial on how to calculate partial derivatives. The ideas are applied to show that certain functions satisfy a famous partial differential equation, known as the wave equation. Such ideas are seen in university mathematics.
This video shows how to calculate partial derivatives via the chain rule. Such ideas are seen in first year university.
This is a basic tutorial on how to calculate a Taylor polynomial for a function of two variables. The ideas are applied to approximate a difficult square root. Such concepts are seen in university mathematics.
Explain the calculus of error estimation with partial derivatives via a simple example. Such ideas are seen in university mathematics.
This presentation shows how to differentiate under integral signs via. Leibniz rule. Many examples are discussed to illustrate the ideas. A proof is also given of the most basic case of Leibniz rule. Such ideas are important in applied mathematics and engineering, for example, in Laplace transforms.
This video shows how to calculate and classify the critical points of functions of two variables. The ideas involve first and second order derivatives and are seen in university mathematics.
This is an example illustrating how to find and classify the critical points of functions of two variables. Such ideas rely on the second derivative test and are seen in university mathematics.
I discuss a basic example of maximizing / minimizing a function subject to a constraint. The approach involves the method of Lagrange multipliers.
A basic review example showing how to use Lagrange multipliers to maximize / minimum a function that is subject to a constraint.
This video shows how to apply the method of Lagrange multipliers to a max/min problem. Such ideas are seen in university mathematics.
An introduction to the calculus of vector functions of one variable. Some simple problems are discussed, including differentiation, integration and how to determine the curve associated with a function (in this example, a helix). Such functions can be used to describe curves and motion in space.
A basic tutorial on the gradient field of a function. We show how to compute the gradient; its geometric significance; and how it is used when computing the directional derivative.The gradient is a basic property of vector calculus.
A basic lecture discussing the divergence of a vector field. I show how to calculate the divergence and present some geometric explanation of what the divergence represents. Several examples are discussed. Such ideas have important applications in fluid flow and are seen in vector calculus.
A basic introduction to the curl of a vector field. I discuss how to calculate the curl and some geometric interpretation. Such ideas are important in fluid flow and are seen in vector calculus.
A basic introduction on how to integrate over curves (line integrals). Several examples are discussed involving scalar functions and vector fields. Such ideas find important applications in engineering and physics.
Basic examples on divergence, curl and line integrals from vector calculus. The examples are discussed and solved as a revision-type question.
An introduction on how to calculate flux via line integrals. Several methods are discussed via examples. Such ideas have important applications to fluid flow and are seen in vector calculus.
Revision question on curl, grad and line integrals. I prove that a given vector field is irrotation and then determine its potential function. The ideas are applied to calculate a line integral via the fundamental theorem of line integrals.
This video shows how to integrate over rectangles. The ideas use double integrals and are seen in university mathematics.
A tutorial on the basics of setting up and evaluating double integrals. We show how to sketch regions of integration, their description, and how to reverse the order of integration.
This video shows how to reverse the order of integration in double integrals. Such ideas can simplify the calculations and are seen in university mathematics.
This video shows how to use double integrals to compute areas of shapes and regions. Such ideas are seen in university mathematics.
How to apply polar coordinates in double integrals for those wanting to review their understanding.
This video shows how to cast and evaluate double integrals in polar co-ordinates. Such ideas are seen in university mathematics.
A basic example of how to calculate the centroid of a region via double integrals. Such problems are seen in university mathematics.
A basic tutorial on how to determine the center of mass of a thin plate. The technique involves a double integral of the density function and is a common problem in applied mathematics, physics and engineering.
A basic introduction on how to solve linear, first-order differential equations. The study of such equations is motivated by their applications to modelling.
I discuss and solve a "homogeneous" first order ordinary differential equation. The method involves a substitution. Such an example is seen in 1st and 2nd year university mathematics.
A basic introduction / revision of how to solve 2nd order homogeneous ordinary differential equations with constant coefficients. Several examples are presented and some applications to vibrating systems are discussed.
A basic lecture showing how to solve nonhomogeneous second-order ordinary differential equations with constant coefficients. The approach illustrated uses the method of undetermined coefficients. I present several examples and show why the method works.
A basic illustration of how to apply the variation of constants / parameters method to solve second order differential equations.
This is a basic introduction to the Laplace transform and how to calculate it. Such ideas are seen in university mathematics.
This video shows how to apply the First shifting theorem of Laplace transforms. Several examples are presented to illustrate how to take the Laplace transform and inverse Laplace transform and are seen in university mathematics.
This video shows how to apply the second shifting theorem of Laplace transforms. Several examples are presented to illustrate how to use the concepts. Such ideas are seen in university mathematics.
How to solve differential equations by the method of Laplace transforms. Such ideas are seen in university mathematics.
This is a basic introduction to Fourier series and how to calculate them. An example is presented that illustrates the computations involved. Such ideas are seen in university mathematics.
How to compute Fourier series of odd and even functions. Several examples are discussed to highlight the ideas.
A review of Fourier series. Several examples are discussed to illustrate the ideas.
This video shows how to solve differential equations via Fourier series. A simple example is presented illustrating the ideas, which are seen in university mathematics.
Derive the heat equation in one dimension. This famous PDE is one of the basic equations from applied mathematics, physics and engineering. This presentation is an introduction to the heat equation.
How solve the heat equation via separation of variables. Such ideas are seen in university mathematics, physics and engineering courses.
How to solve the heat equation via separation of variables and Fourier series.
How to solve the wave equation via Fourier series and separation of variables. Such ideas are have important applications in science, engineering and physics.