Coordinate Systems for Multiple Integration

Introduction

It is sometimes nessecary to use different coordinate systems to evaluate integrals, especially when dealing with complex geometries. The most common coordinate systems used in Calc III are Cartesian, Polar, Cylindrical, and Spherical coordinates. Each of these coordinate systems has its own advantages and is suited for different types of problems.

We will also introduce the general idea of Jacobians, which are used to transform integrals from one coordinate system to another.

Polar and Cylindrical Coordinates Coordinates

Polar coordinates are a two-dimensional coordinate system where each point in the plane is determined by a radial distance $r$ from a fixed point (typically the origin) and an angle $\theta$ from a fixed direction (typically from the positive x-axis). They are particulately useful for calculating double integrals over circular or sector-shaped regions.

General Formulas

The transformation from Cartesian coordinates $(x, y)$ to Polar coordinates $(r, \theta)$ is given by:

When changing variables in a double integral from Cartesian to Polar coordinates, the area element $dA$ transforms as follows:

Please note the extra factor of $r$ in the area element. This accounts for the stretching of the area in Polar coordinates, and is a special case of the Jacobian determinant for this transformation. We will discuss Jacobians in more detail later.

Cylindrical coordinates are a three-dimensional coordinate system that extends Polar coordinates by adding a height component $z$. A point in space is represented by the tuplet $(r, \theta, z)$, where $r$ and $\theta$ are the same as in Polar coordinates, and $z$ is the height above the xy-plane. Cylindrical coordinates are particularly useful for problems with symmetry around an axis, such as cylinders and cones.

General Formulas

The transformation from Cartesian coordinates $(x, y, z)$ to Cylindrical coordinates $(r, \theta, z)$ is given by:

When changing variables in a triple integral from Cartesian to Cylindrical coordinates, the volume element $dV$ transforms as follows:

Again, note the extra factor of $r$ in the volume element, which accounts for the geometry of the transformation.. There is no additional factor becayse the height $z$ remains unchanged.

Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system where each point in space is determined by three parameters: the radial distance $\rho$ from the origin, the polar angle $\phi$ (the angle from the positive z-axis), and the azimuthal angle $\theta$ (the angle from the positive x-axis in the xy-plane). Spherical coordinates are particularly useful for problems with spherical symmetry, such as spheres and shells.

Please note that the notation for spherical coordinates can vary; some texts use $r$ instead of $\rho$ for the radial distance. Additionally, some texts may use different conventions for the angles $\phi$ and $\theta$. Why you might ask? Because mathematicians, physicists, and engineers clearly hate each other and like to keep things complicated.

”Mathematician” Convention

”Physics and Engineering” Convention

You might ask: What is point? Will these result in the exact same calculation? Yes. Yes they will.

We will use the “mathematician” convention for the rest of this section, simply because it was popularized by Stewart’s Calculus textbooks.

When changing variables in a triple integral from Cartesian to Spherical coordinates, the volume element $dV$ transforms as follows:

Please note the extra factors of $\rho^2 \sin \phi$ in the volume element, which account for the geometry of the transformation.

Jacobians and General Change of Variables

Remember U-Substitution from single variable calculus? Jacobians are the multi-variable extension of that idea. When changing variables in multiple integrals, we need to account for how the area or volume elements change under the transformation. This is done using the Jacobian determinant.

Definition of the Jacobian

Suppose we have a transformation from variables $(u, v)$ to variables $(x, y)$ given by:

Provided the transformation is differentiable and has a non-zero Jacobian determinant, the area element transforms as follows:

where the Jacobian determinant is given by:

Then, the double integral transforms as follows:

where $R$ is the region in the xy-plane and $S$ is the corresponding region in the uv-plane.

Similarly, for a transformation from variables $(u, v, w)$ to variables $(x, y, z)$ given by:

Provided the transformation is differentiable and has a non-zero Jacobian determinant, the volume element transforms as follows:

where the Jacobian determinant is given by:

Then, the triple integral transforms as follows:

where $R$ is the region in the xyz-space and $S$ is the corresponding region in the uvw-space.