Coordinate Systems

Comprehensive practice questions covering Cartesian, Cylindrical, and Spherical coordinate systems, including basis vectors, elementary displacements, surfaces, and volume calculations based on the lecture transcript.

How is a coordinate system defined in the lecture introduction?

A mathematical system used to describe the position of a point in space, represented in 3D using 3 axes perpendicular to each other.

What is the role of the index $i$ in the notation for basis vectors $\vec{e}_i$?

The index $i$ indicates the direction of the unit vector (which is a synonym for basis vector).

In Cartesian Coordinates, what does it mean for the basis vectors $(\vec{e}_x, \vec{e}_y, \vec{e}_z)$ to be an orthonormal basis?

It means that any vector in space can be expressed as a linear combination of the basis vectors, which are unit vectors: $\vec{e}_x = (1, 0, 0)$, $\vec{e}_y = (0, 1, 0)$, and $\vec{e}_z = (0, 0, 1)$.

How is a unit vector $\vec{e}_x$ in the direction of vector $\vec{AB}$ calculated?

It is obtained by dividing $\vec{AB}$ by its magnitude $\Vert\vec{AB}\Vert$: $\vec{e}_x = \frac{\vec{AB}}{\Vert\vec{AB}\Vert}$.

What is the expression for the position vector $\vec{OM}$ of point $M(x, y, z)$ in Cartesian coordinates?

$$\vec{OM} = x \vec{e}_x + y \vec{e}_y + z \vec{e}_z$$

What is the expression for an elementary displacement vector $\vec{dl}$ in Cartesian coordinates?

$$\vec{dl} = dx \vec{e}_x + dy \vec{e}_y + dz \vec{e}_z$$

What is the expression for the elementary volume $dV$ in Cartesian coordinates?

$$dV = dx dy dz$$

Which three coordinates define the position of a point $M$ in the cylindrical coordinate system?

The radial distance $r$, the angle $\theta$, and the vertical coordinate $z$.

Why are the basis vectors $\vec{e}<em>r$, $\vec{e}</em>\theta$, and $\vec{e}_z$ in cylindrical coordinates considered "mobile"?

Unlike Cartesian basis vectors, they are not fixed in space; $\vec{e}<em>r$ and $\vec{e}</em>\theta$ change direction as the point $M$ moves.

How is the position vector $\vec{OM}$ represented in cylindrical coordinates?

$$\vec{OM} = r \vec{e}_r + z \vec{e}_z$$

What are the projections of the cylindrical basis vectors $\vec{e}<em>r$ and $\vec{e}</em>\theta$ onto the Cartesian axes $\vec{e}_x$ and $\vec{e}_y$?

$\vec{e}<em>r = \cos(\theta) \vec{e}_x + \sin(\theta) \vec{e}_y$ and $\vec{e}</em>\theta = -\sin(\theta) \vec{e}_x + \cos(\theta) \vec{e}_y$.

What is the relation for the elementary displacement $d\vec{OM}$ in cylindrical coordinates?

$$d\vec{OM} = dr \vec{e}<em>r + r d\theta \vec{e}</em>\theta + dz \vec{e}_z$$

What is the elementary volume element $dV$ in cylindrical coordinates?

$$dV = r dr d\theta dz$$

What are the typical integration limits for the total volume integral in cylindrical coordinates?

$\theta$ between $0$ and $2\pi$, $r$ between $0$ and $R$, and $z$ between $0$ and $h$.

What is the formula for a surface element $d\vec{S}$ normal to the $z$-axis in cylindrical coordinates?

$$d\vec{S} = r dr d\theta \vec{e}_z$$

Describe the three coordinates used in the spherical coordinate system: $r$, $\theta$, and $\phi$.

Coordinate $r$ is the radial distance $OM$, $\theta$ is the polar angle ($0$ to $\pi$), and $\phi$ is the azimuthal angle in the $P(xy)$ plane ($0$ to $2\pi$).

What is the relationship between Cartesian coordinates and spherical coordinates $(r, \theta, \phi)$?

$x = r \cos(\phi) \sin(\theta)$, $y = r \sin(\phi) \sin(\theta)$, and $z = r \cos(\theta)$.

What is the expression for the elementary volume $dV$ in spherical coordinates?

$$dV = r^2 \sin(\theta) dr d\theta d\phi$$

What is the differential surface element $d\vec{S}$ normal to the $r$ direction in spherical coordinates?

$$d\vec{S} = r^2 \sin(\theta) d\theta d\phi \vec{e}_r$$

Using spherical coordinates, show the final formula for the volume of a sphere of radius $R$.

$$V = \frac{4\pi R^3}{3}$$

What is the formula for the volume of a cylinder with radius $R$ and height $h$ as derived in the notes?

$$V = \pi R^2 h$$

How is the total volume of a parallelepiped with sides $a$, $b$, and $c$ calculated in Cartesian coordinates?

$$V = \int_0^a dx \int_0^b dy \int_0^c dz = abc$$