
Indefinite Integration
The following quote from Reference [1] can clarify a lot of confusion about indefinite integration: Today the process of finding the fluent of a given fluxion is called indefinite integration, or antidifferentiation, and the result of integrating a given function is its indefinite integral, or antiderivative (the “indefinite” refers to the existence of the arbitrary…

Multiple Cycles of the Complex Exponential Function
A Definite Integral of the Complex Exponential Function Recall from this post that: $ \int_0^{2 \pi} d\theta \cos\theta + i \int_0^{2 \pi} d\theta \sin\theta = 0$ and $ \int_0^{2 \pi} d\theta e^{i \theta} = 0.$ These are definite integrals because the upper and lower limits of integration are finite numbers. Symmetry of the Sine and Cosine Functions…

How to Integrate in a Spherical Coordinate System
Review of Integration Integration with Cartesian coordinates is simple. The general form is $\int\int\int f(x,y,z)dxdydz$ in which $f(x,y,z)$ is an arbitrary function of the Cartesian coordinates. However, there may be cases in which integrating with spherical coordinates is more convenient. Given the above, general form for integration with Cartesian coordinates, how can one integrate in a spherical…

The Crux of Calculus
Define $\Delta x \equiv x_2 – x_1$, to be consistent with this post. Similarly, define $\Delta y \equiv y_2 – y_1$ and $\Delta z \equiv z_2 – z_1$. The Cartesian coordinates are $x$, $y$, & $z$. In contrast, the spherical coordinates are $r$, $\theta$, & $\phi$. Here, $\phi$ is the azimuthal angle in the $xy$plane. Next,…