# Problems

• • Contributed by Marcia Levitus

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

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