
How a Multivariable Function Changes with Respect to the Cartesian Coordinates
Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. This post shows how to calculate a partial derivative of a multivariable function $F(x,y,z)$ with respect to each Cartesian coordinates. The resulting expressions are in spherical coordinates. In previous posts (see references [2], [3], and [4]), the following equations were written: $ \frac{\partial }{\partial r} F(x,y,z) = \bigg( ( \frac{\partial x}{\partial r}…

The Polar Unit Vector
Consider a spherical coordinate system. Let a point be represented by $(r, \theta, \phi)$, in that order. Now that the order of the coordinates is established, I can define unit vectors that form a righthanded coordinate system. Suppose the radial unit vector $\vec{e}_r$ points radially outward from the origin to the point, and the polar…

Defining Del in Terms of Cartesian Coordinates
What is Del? In math, the symbol $\vec{\nabla}$ is called “del.” This symbol is defined in terms of Cartesian coordinates. $\vec{\nabla} := \vec{e}_x\frac{\partial}{\partial x} + \vec{e}_y\frac{\partial}{\partial y} + \vec{e}_z\frac{\partial}{\partial z}$ The right side is a sum of unit vectors. So $\vec{\nabla}$ is a vector. This is why I write $\vec{\nabla}$ instead of just $\nabla$.In the…

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…