# 13.5: Problems

- Page ID
- 106886

Problem \(\PageIndex{1}\)

Finish the problem of Example \(13.1.1\) and obtain \(y\) and \(z\).

Problem \(\PageIndex{2}\)

Use determinants to solve the equations:

A)

\[\begin{array}{c} x+y+z=6\\ x+2y+3z=14\\ x+4y+9z=36 \end{array} \nonumber\]

B)

\[ \begin{array}{c} x+iy-z=0\\ ix+y+z=0\\ x+2y-iz=1 \end{array} \nonumber\]

Problem \(\PageIndex{3}\)

Show that a \(3\times 3\) determinant that contains zeros above the principal diagonal is the product of the diagonal elements.

\[D=\begin{vmatrix} a &0&0 \\ b&c &0 \\ d& e &f \end{vmatrix}=acf \nonumber\]

Problem \(\PageIndex{4}\)

Prove that

\[D=\begin{vmatrix} 1 &2&3 \\ 2&3 &3 \\ 3& 4 &3 \end{vmatrix}=0 \nonumber\]

using the properties of determinants (that is, without calculating the determinant!). Clearly state the properties you use in each step.

Exercise \(\PageIndex{5}\)

In previous lectures, we discussed how to perform double and triple integrals in different coordinate systems. For instance, we learned that the area elements and volume elements are:

2D:

Cartesian: \(dA= dx.dy\)

Polar: \(dA=r. dr. d\theta\)

3D:

Cartesian: \(dV= dx.dy.dz\)

Spherical: \(dV=r^2.\sin\theta dr. d\theta d\phi\)

In general, for any coordinate system, we can express the area (or volume) element in a new coordinate system using the Jacobian (\(J\)). For example, in polar coordinates in two dimensions:

\[dA=dx.dy=J. dr.d\theta \nonumber\]

where the Jacobian is defined as:

\[J=\left | \begin{matrix} \frac{\partial x}{\partial r} &\frac{\partial x}{\partial \theta} \\ \\ \frac{\partial y}{\partial r} &\frac{\partial y}{\partial \theta} \end{matrix} \right | \nonumber\]

a) Calculate the Jacobian in two-dimensional polar coordinates and show that \(dA=r. dr. d\theta\).

In spherical coordinates,

\[dV=dx.dy.dz=J. dr.d\theta. d\phi \nonumber\]

where

\[J=\left | \begin{matrix} \frac{\partial x}{\partial r} &\frac{\partial x}{\partial \theta}&\frac{\partial x}{\partial \phi} \\ \\ \frac{\partial y}{\partial r} &\frac{\partial y}{\partial \theta}&\frac{\partial y}{\partial \phi}\\ \\ \frac{\partial z}{\partial r} &\frac{\partial z}{\partial \theta}&\frac{\partial z}{\partial \phi}\\ \end{matrix} \right | \nonumber\]

b) Calculate the Jacobian in three-dimensional spherical coordinates and show that

\[dV=r^2.\sin\theta dr. d\theta d\phi \nonumber\]