The Central Idea
A point in space can be described in more than one way.
The same physical location may be written using:
- Cartesian coordinates \((x,y,z)\);
- cylindrical coordinates \((r,\theta,z)\);
- spherical coordinates \((\rho,\theta,\phi)\).
The coordinate system is a language for geometry.
A good coordinate choice can turn a difficult vector-calculus problem into a simple one.
Coordinates Versus Geometry
The vector field itself is a geometric object.
Its coordinate components depend on the coordinate system used to describe it.
The same field may have a simple Cartesian formula and a complicated spherical formula, or the reverse.
Why Curved Coordinates Change the Formulas
In Cartesian coordinates, the basis vectors
\[ \mathbf{i}, \qquad \mathbf{j}, \qquad \mathbf{k} \]are constant everywhere.
In cylindrical and spherical coordinates, some basis directions rotate as position changes.
In addition, equal coordinate increments do not always represent equal physical distances.
Cartesian Coordinates
Cartesian coordinates describe position using perpendicular distances along fixed axes:
\[ \boxed{ (x,y,z) }. \]The position vector is
\[ \boxed{ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} }. \]Cartesian Basis Vectors
\[ \mathbf{i} = \langle1,0,0\rangle, \qquad \mathbf{j} = \langle0,1,0\rangle, \qquad \mathbf{k} = \langle0,0,1\rangle. \]These unit vectors do not depend on position.
Cartesian Differential Length
A small displacement is
\[ \boxed{ d\mathbf{r} = \mathbf{i}\,dx + \mathbf{j}\,dy + \mathbf{k}\,dz }. \]The squared distance element is
\[ \boxed{ ds^2 = dx^2+dy^2+dz^2 }. \]Cartesian Volume Element
A small Cartesian box has volume
\[ \boxed{ dV = dx\,dy\,dz }. \]No additional geometric scale factor is required.
Cartesian Surface Elements
On a coordinate surface:
\[ x=\text{constant} \quad\Rightarrow\quad dS=dy\,dz, \] \[ y=\text{constant} \quad\Rightarrow\quad dS=dx\,dz, \] \[ z=\text{constant} \quad\Rightarrow\quad dS=dx\,dy. \]Cartesian Gradient
For a scalar field \(f(x,y,z)\),
\[ \boxed{ \nabla f = \mathbf{i}\frac{\partial f}{\partial x} + \mathbf{j}\frac{\partial f}{\partial y} + \mathbf{k}\frac{\partial f}{\partial z} }. \]Cartesian Divergence
For
\[ \mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}, \]divergence is
\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} }. \]Cartesian Curl
Curl is
\[ \boxed{ \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z}\\ F_x & F_y & F_z \end{vmatrix} }. \]When Cartesian Coordinates Are Natural
Cartesian coordinates are usually preferred for:
- rectangular boxes;
- planes;
- coordinate-aligned boundaries;
- polynomial fields in \(x,y,z\);
- problems with rectangular symmetry.
Cylindrical Coordinates
Cylindrical coordinates extend polar coordinates into three dimensions:
\[ \boxed{ (r,\theta,z) }. \]Here:
- \(r\) is distance from the \(z\)-axis;
- \(\theta\) is angle around the \(z\)-axis;
- \(z\) is the usual vertical coordinate.
Cylindrical-to-Cartesian Conversion
\[ \boxed{ x=r\cos\theta } \] \[ \boxed{ y=r\sin\theta } \] \[ \boxed{ z=z }. \]Cartesian-to-Cylindrical Conversion
\[ \boxed{ r=\sqrt{x^2+y^2} } \] \[ \boxed{ \theta=\operatorname{atan2}(y,x) } \] \[ \boxed{ z=z }. \]The angle must be placed in the correct quadrant.
Cylindrical Coordinate Surfaces
Holding one cylindrical coordinate constant produces familiar surfaces.
If
\[ r=R, \]we obtain a cylinder of radius \(R\).
If
\[ \theta=\theta_0, \]we obtain a vertical half-plane through the \(z\)-axis.
If
\[ z=z_0, \]we obtain a horizontal plane.
Cylindrical Basis Vectors
The unit vectors are
\[ \boxed{ \mathbf{e}_r = \langle\cos\theta,\sin\theta,0\rangle } \] \[ \boxed{ \mathbf{e}_\theta = \langle-\sin\theta,\cos\theta,0\rangle } \] \[ \boxed{ \mathbf{e}_z = \mathbf{k} }. \]The Cylindrical Basis Rotates
Unlike \(\mathbf{i}\) and \(\mathbf{j}\),
\[ \mathbf{e}_r \]and
\[ \mathbf{e}_\theta \]depend on \(\theta\).
As we move around the \(z\)-axis, the basis itself rotates.
Derivatives of Cylindrical Basis Vectors
\[ \boxed{ \frac{\partial\mathbf{e}_r}{\partial\theta} = \mathbf{e}_\theta } \] \[ \boxed{ \frac{\partial\mathbf{e}_\theta}{\partial\theta} = -\mathbf{e}_r }. \]These changing basis vectors are one reason vector-calculus formulas gain extra terms in curved coordinates.
Cylindrical Position Vector
The position vector can be written
\[ \boxed{ \mathbf{r} = r\mathbf{e}_r + z\mathbf{e}_z }. \]There is no separate \(\theta\mathbf{e}_\theta\) term because \(\theta\) describes direction rather than distance from the axis.
Cylindrical Differential Length
A small displacement is
\[ \boxed{ d\mathbf{r} = \mathbf{e}_r\,dr + \mathbf{e}_\theta\,r\,d\theta + \mathbf{e}_z\,dz }. \]Therefore,
\[ \boxed{ ds^2 = dr^2+r^2d\theta^2+dz^2 }. \]Why the Factor \(r\) Appears
A small angular change \(d\theta\) at radius \(r\) sweeps arc length
\[ \boxed{ r\,d\theta }. \]The same angular change produces a larger physical displacement farther from the axis.
Cylindrical Scale Factors
Cylindrical coordinates have scale factors
\[ \boxed{ h_r=1, \qquad h_\theta=r, \qquad h_z=1 }. \]These factors encode how coordinate increments convert to physical lengths.
Cylindrical Volume Element
Multiply the three scale factors:
\[ h_rh_\theta h_z = r. \]Therefore,
\[ \boxed{ dV = r\,dr\,d\theta\,dz }. \]Cylindrical Coordinate Surface Elements
For
\[ r=\text{constant}, \]the surface element is
\[ \boxed{ dS = r\,d\theta\,dz }. \]For
\[ \theta=\text{constant}, \]it is
\[ \boxed{ dS = dr\,dz }. \]For
\[ z=\text{constant}, \]it is
\[ \boxed{ dS = r\,dr\,d\theta }. \]Cylindrical Gradient
For \(f(r,\theta,z)\),
\[ \boxed{ \nabla f = \mathbf{e}_r \frac{\partial f}{\partial r} + \mathbf{e}_\theta \frac{1}{r} \frac{\partial f}{\partial\theta} + \mathbf{e}_z \frac{\partial f}{\partial z} }. \]Why the Angular Gradient Component Has \(1/r\)
The derivative
\[ \frac{\partial f}{\partial\theta} \]measures change per unit angle.
The physical distance associated with \(d\theta\) is
\[ r\,d\theta. \]Therefore, change per unit distance requires division by \(r\).
Cylindrical Divergence
If
\[ \mathbf{F} = F_r\mathbf{e}_r + F_\theta\mathbf{e}_\theta + F_z\mathbf{e}_z, \]then
\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} (rF_r) + \frac{1}{r} \frac{\partial F_\theta}{\partial\theta} + \frac{\partial F_z}{\partial z} }. \]Cylindrical Curl
The curl is
\[ \boxed{ \nabla\times\mathbf{F} = \mathbf{e}_r \left( \frac{1}{r}\frac{\partial F_z}{\partial\theta} - \frac{\partial F_\theta}{\partial z} \right) + \mathbf{e}_\theta \left( \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r} \right) + \mathbf{e}_z \frac{1}{r} \left( \frac{\partial}{\partial r}(rF_\theta) - \frac{\partial F_r}{\partial\theta} \right) }. \]Cylindrical Laplacian of a Scalar Field
\[ \boxed{ \nabla^2f = \frac{1}{r} \frac{\partial}{\partial r} \left( r\frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2f}{\partial\theta^2} + \frac{\partial^2f}{\partial z^2} }. \]When Cylindrical Coordinates Are Natural
Cylindrical coordinates are especially useful for:
- cylinders;
- pipes;
- axial symmetry;
- rotation around the \(z\)-axis;
- fields depending mainly on distance from an axis;
- helical or circular geometries.
Example: Radial Cylindrical Field
Let
\[ \mathbf{F} = r\mathbf{e}_r. \]Then
\[ F_r=r, \qquad F_\theta=0, \qquad F_z=0. \]Therefore,
\[ \nabla\cdot\mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} (r^2) = \frac{1}{r}(2r) = \boxed{2}. \]Spherical Coordinates
Spherical coordinates describe position using:
\[ \boxed{ (\rho,\theta,\phi) }. \]Here:
- \(\rho\) is distance from the origin;
- \(\theta\) is the azimuthal angle around the \(z\)-axis;
- \(\phi\) is the polar angle measured downward from the positive \(z\)-axis.
Spherical-to-Cartesian Conversion
\[ \boxed{ x = \rho\sin\phi\cos\theta } \] \[ \boxed{ y = \rho\sin\phi\sin\theta } \] \[ \boxed{ z = \rho\cos\phi }. \]Cartesian-to-Spherical Conversion
\[ \boxed{ \rho = \sqrt{x^2+y^2+z^2} } \] \[ \boxed{ \theta = \operatorname{atan2}(y,x) } \] \[ \boxed{ \phi = \arccos\left(\frac{z}{\rho}\right) }. \]Spherical Coordinate Surfaces
If
\[ \rho=R, \]we obtain a sphere of radius \(R\).
If
\[ \theta=\theta_0, \]we obtain a vertical half-plane through the \(z\)-axis.
If
\[ \phi=\phi_0, \]we obtain a cone with vertex at the origin.
Spherical Basis Vectors
The spherical unit vectors are
\[ \boxed{ \mathbf{e}_\rho = \langle \sin\phi\cos\theta, \sin\phi\sin\theta, \cos\phi \rangle } \] \[ \boxed{ \mathbf{e}_\theta = \langle -\sin\theta, \cos\theta, 0 \rangle } \] \[ \boxed{ \mathbf{e}_\phi = \langle \cos\phi\cos\theta, \cos\phi\sin\theta, -\sin\phi \rangle }. \]All Spherical Basis Directions Vary
In spherical coordinates, every basis vector depends on position through the angles.
This makes spherical vector-calculus formulas more elaborate than their Cartesian counterparts.
Spherical Position Vector
The position vector is especially simple:
\[ \boxed{ \mathbf{r} = \rho\mathbf{e}_\rho }. \]The radial basis vector points directly away from the origin.
Spherical Differential Length
A small displacement is
\[ \boxed{ d\mathbf{r} = \mathbf{e}_\rho\,d\rho + \mathbf{e}_\theta\,\rho\sin\phi\,d\theta + \mathbf{e}_\phi\,\rho\,d\phi }. \]Therefore,
\[ \boxed{ ds^2 = d\rho^2 + \rho^2\sin^2\phi\,d\theta^2 + \rho^2\,d\phi^2 }. \]Spherical Scale Factors
The scale factors are
\[ \boxed{ h_\rho=1, \qquad h_\theta=\rho\sin\phi, \qquad h_\phi=\rho }. \]Spherical Volume Element
Multiplying the scale factors gives
\[ \rho^2\sin\phi. \]Therefore,
\[ \boxed{ dV = \rho^2\sin\phi \,d\rho\,d\theta\,d\phi }. \]Spherical Coordinate Surface Elements
For
\[ \rho=\text{constant}, \]the surface element is
\[ \boxed{ dS = \rho^2\sin\phi \,d\theta\,d\phi }. \]For
\[ \theta=\text{constant}, \]the surface element is
\[ \boxed{ dS = \rho\,d\rho\,d\phi }. \]For
\[ \phi=\text{constant}, \]it is
\[ \boxed{ dS = \rho\sin\phi \,d\rho\,d\theta }. \]Spherical Gradient
For \(f(\rho,\theta,\phi)\),
\[ \boxed{ \nabla f = \mathbf{e}_\rho \frac{\partial f}{\partial\rho} + \mathbf{e}_\theta \frac{1}{\rho\sin\phi} \frac{\partial f}{\partial\theta} + \mathbf{e}_\phi \frac{1}{\rho} \frac{\partial f}{\partial\phi} }. \]Spherical Divergence
If
\[ \mathbf{F} = F_\rho\mathbf{e}_\rho + F_\theta\mathbf{e}_\theta + F_\phi\mathbf{e}_\phi, \]then
\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{1}{\rho^2} \frac{\partial}{\partial\rho} \left( \rho^2F_\rho \right) + \frac{1}{\rho\sin\phi} \frac{\partial F_\theta}{\partial\theta} + \frac{1}{\rho\sin\phi} \frac{\partial}{\partial\phi} \left( \sin\phi\,F_\phi \right) }. \]Spherical Curl
The curl components are
\[ \boxed{ (\nabla\times\mathbf{F})_\rho = \frac{1}{\rho\sin\phi} \left[ \frac{\partial}{\partial\theta} (\sin\phi\,F_\phi) - \frac{\partial F_\theta}{\partial\phi} \right] } \] \[ \boxed{ (\nabla\times\mathbf{F})_\theta = \frac{1}{\rho} \left[ \frac{\partial F_\rho}{\partial\phi} - \frac{\partial}{\partial\rho} (\rho F_\phi) \right] } \] \[ \boxed{ (\nabla\times\mathbf{F})_\phi = \frac{1}{\rho} \left[ \frac{1}{\sin\phi} \frac{\partial F_\rho}{\partial\theta} - \frac{\partial}{\partial\rho} (\rho F_\theta) \right] }. \]The full curl is
\[ \nabla\times\mathbf{F} = (\nabla\times\mathbf{F})_\rho\mathbf{e}_\rho + (\nabla\times\mathbf{F})_\theta\mathbf{e}_\theta + (\nabla\times\mathbf{F})_\phi\mathbf{e}_\phi. \]Spherical Laplacian of a Scalar Field
\[ \boxed{ \nabla^2f = \frac{1}{\rho^2} \frac{\partial}{\partial\rho} \left( \rho^2 \frac{\partial f}{\partial\rho} \right) + \frac{1}{\rho^2\sin^2\phi} \frac{\partial^2f}{\partial\theta^2} + \frac{1}{\rho^2\sin\phi} \frac{\partial}{\partial\phi} \left( \sin\phi \frac{\partial f}{\partial\phi} \right) }. \]When Spherical Coordinates Are Natural
Spherical coordinates are especially useful for:
- spheres and balls;
- radial fields;
- inverse-square models;
- central-force problems;
- point-source fields;
- problems with full rotational symmetry about the origin.
Example: Pure Radial Spherical Field
Let
\[ \mathbf{F} = \rho\,\mathbf{e}_\rho. \]Then
\[ F_\rho=\rho, \qquad F_\theta=0, \qquad F_\phi=0. \]Therefore,
\[ \nabla\cdot\mathbf{F} = \frac{1}{\rho^2} \frac{\partial}{\partial\rho} (\rho^3) = \frac{3\rho^2}{\rho^2} = \boxed{3}. \]Inverse-Square Radial Field
Consider
\[ \boxed{ \mathbf{F} = \frac{C}{\rho^2} \mathbf{e}_\rho }. \]Away from the origin,
\[ \nabla\cdot\mathbf{F} = \frac{1}{\rho^2} \frac{\partial}{\partial\rho} \left( \rho^2 \frac{C}{\rho^2} \right) = \frac{1}{\rho^2} \frac{\partial C}{\partial\rho} = 0. \]The origin is singular and requires separate attention.
Why Inverse-Square Flux Is Constant
On a sphere of radius \(\rho\),
\[ \mathbf{F}\cdot\mathbf{n} = \frac{C}{\rho^2}. \]The sphere area is
\[ 4\pi\rho^2. \]Therefore,
\[ \boxed{ \Phi = 4\pi C }. \]The \(\rho^2\) in area exactly cancels the \(1/\rho^2\) field strength.
Orthogonal Curvilinear Coordinates
Cartesian, cylindrical, and spherical coordinates are all orthogonal coordinate systems.
Their local basis directions are mutually perpendicular.
What changes from one system to another are the scale factors.
General Scale-Factor Pattern
For orthogonal coordinates
\[ (q_1,q_2,q_3) \]with scale factors
\[ h_1,h_2,h_3, \]the differential length is
\[ \boxed{ d\mathbf{r} = \mathbf{e}_1h_1\,dq_1 + \mathbf{e}_2h_2\,dq_2 + \mathbf{e}_3h_3\,dq_3 }. \]General Volume Element
The orthogonal-coordinate volume element is
\[ \boxed{ dV = h_1h_2h_3 \,dq_1\,dq_2\,dq_3 }. \]The Jacobian factors in cylindrical and spherical coordinates are examples of this rule.
General Gradient Pattern
For an orthogonal coordinate system,
\[ \boxed{ \nabla f = \sum_{i=1}^{3} \mathbf{e}_i \frac{1}{h_i} \frac{\partial f}{\partial q_i} }. \]This reveals why scale factors divide directional derivatives.
General Divergence Pattern
If
\[ \mathbf{F} = F_1\mathbf{e}_1 + F_2\mathbf{e}_2 + F_3\mathbf{e}_3, \]then
\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{1}{h_1h_2h_3} \left[ \frac{\partial}{\partial q_1} (h_2h_3F_1) + \frac{\partial}{\partial q_2} (h_3h_1F_2) + \frac{\partial}{\partial q_3} (h_1h_2F_3) \right] }. \]Coordinate Singularities
Curved coordinate systems can become singular even when the underlying geometry is perfectly regular.
In cylindrical coordinates, \(r=0\) makes the angle \(\theta\) undefined.
In spherical coordinates, \(\rho=0\) makes both angles undefined, and the poles create angular degeneracy.
Coordinate Singularity Versus Physical Singularity
A coordinate singularity is a failure of the coordinate description.
A physical or field singularity is a failure of the field itself to remain finite or defined.
These are not the same thing.
Example: The Origin in Spherical Coordinates
The origin has
\[ \rho=0. \]At that point, infinitely many angular pairs
\[ (\theta,\phi) \]represent the same location.
This is a coordinate degeneracy, not necessarily a singularity in the physical field.
Changing Vector Components
A vector can be expressed in different bases.
In the plane,
\[ \mathbf{i} = \cos\theta\,\mathbf{e}_r - \sin\theta\,\mathbf{e}_\theta \]and
\[ \mathbf{j} = \sin\theta\,\mathbf{e}_r + \cos\theta\,\mathbf{e}_\theta. \]Cartesian to Cylindrical Vector Components
For
\[ \mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}, \]cylindrical components are
\[ \boxed{ F_r = F_x\cos\theta+F_y\sin\theta } \] \[ \boxed{ F_\theta = -F_x\sin\theta+F_y\cos\theta } \] \[ \boxed{ F_z=F_z }. \]Example: Convert \(\langle x,y,0\rangle\)
Since
\[ x=r\cos\theta, \qquad y=r\sin\theta, \]the Cartesian field
\[ \mathbf{F} = \langle x,y,0\rangle \]becomes
\[ \boxed{ \mathbf{F} = r\mathbf{e}_r }. \]Cylindrical coordinates reveal the radial structure immediately.
Example: Convert a Rotational Field
The Cartesian field
\[ \mathbf{F} = \langle-y,x,0\rangle \]becomes
\[ \boxed{ \mathbf{F} = r\mathbf{e}_\theta }. \]Cylindrical coordinates reveal that the field is purely tangential.
Coordinate Choice and Symmetry
A coordinate system is most effective when coordinate surfaces align with the symmetry of the problem.
If the field is unchanged by rotation around the \(z\)-axis, cylindrical coordinates are often natural.
If the field depends only on distance from the origin, spherical coordinates are often natural.
Recognizing Cylindrical Symmetry
Clues include expressions such as
\[ x^2+y^2 \]or
\[ \sqrt{x^2+y^2}. \]These naturally become
\[ r^2 \]and
\[ r. \]Recognizing Spherical Symmetry
Clues include
\[ x^2+y^2+z^2 \]or
\[ \sqrt{x^2+y^2+z^2}. \]These naturally become
\[ \rho^2 \]and
\[ \rho. \]Coordinate Choice for Line Integrals
A circular path may be easiest in cylindrical coordinates because
\[ r=R \]is constant.
Then
\[ d\mathbf{r} = \mathbf{e}_\theta R\,d\theta \]along a horizontal circle.
Coordinate Choice for Surface Integrals
A cylindrical side surface
\[ r=R \]naturally uses
\[ dS = R\,d\theta\,dz. \]A sphere
\[ \rho=R \]naturally uses
\[ dS = R^2\sin\phi\,d\theta\,d\phi. \]Coordinate Choice for Flux
On
\[ r=R, \]the outward normal is
\[ \mathbf{e}_r. \]On
\[ \rho=R, \]the outward normal is
\[ \mathbf{e}_\rho. \]This alignment can make flux integrals almost immediate.
Coordinate Choice for the Divergence Theorem
The theorem converts flux through a closed surface into a volume integral.
Match the coordinates to the enclosed volume:
- boxes \(\rightarrow\) Cartesian;
- cylinders \(\rightarrow\) cylindrical;
- balls and spherical shells \(\rightarrow\) spherical.
Example: Flux Through a Sphere in Spherical Coordinates
Let
\[ \mathbf{F} = \rho\mathbf{e}_\rho \]on the sphere
\[ \rho=R. \]The outward normal is
\[ \mathbf{e}_\rho. \]Therefore,
\[ \mathbf{F}\cdot\mathbf{n}=R. \]Also,
\[ dS = R^2\sin\phi\,d\theta\,d\phi. \]Hence,
\[ \Phi = \int_0^{2\pi} \int_0^\pi R^3\sin\phi \,d\phi\,d\theta = \boxed{4\pi R^3}. \]Example: Circulation Around a Circle in Cylindrical Coordinates
Let
\[ \mathbf{F} = r\mathbf{e}_\theta \]and let \(C\) be the circle
\[ r=R, \qquad z=0, \]oriented in increasing \(\theta\).
Along the circle,
\[ d\mathbf{r} = \mathbf{e}_\theta R\,d\theta. \]Therefore,
\[ \mathbf{F}\cdot d\mathbf{r} = R^2\,d\theta. \]Thus,
\[ \boxed{ \oint_C \mathbf{F}\cdot d\mathbf{r} = 2\pi R^2 }. \]Common Error: Forgetting the Jacobian
In cylindrical coordinates,
\[ \boxed{ dV=r\,dr\,d\theta\,dz }. \]In spherical coordinates,
\[ \boxed{ dV=\rho^2\sin\phi\,d\rho\,d\theta\,d\phi }. \]Omitting these factors changes the measure.
Common Error: Treating Curved Basis Vectors as Constant
In cylindrical coordinates,
\[ \mathbf{e}_r \]and
\[ \mathbf{e}_\theta \]vary with \(\theta\).
In spherical coordinates, all three basis directions vary.
This is why Cartesian derivative rules cannot simply be copied component by component into curved coordinates.
Common Error: Mixing Spherical Angle Conventions
Different books sometimes interchange the roles of \(\theta\) and \(\phi\).
In this lesson:
\[ \boxed{ \theta = \text{azimuthal angle around the }z\text{-axis} } \]and
\[ \boxed{ \phi = \text{polar angle measured from the positive }z\text{-axis} }. \]Always check the convention before using a formula.
Common Error: Using \(r\) and \(\rho\) Interchangeably
In cylindrical coordinates,
\[ r=\sqrt{x^2+y^2} \]measures distance from the \(z\)-axis.
In spherical coordinates,
\[ \rho=\sqrt{x^2+y^2+z^2} \]measures distance from the origin.
Common Error: Converting Coordinates but Not Vector Components
Replacing
\[ x=r\cos\theta, \qquad y=r\sin\theta \]changes the scalar variables, but a vector field may also need to be rewritten in the cylindrical basis.
The basis conversion is part of the transformation.
Common Error: Choosing a Complicated Coordinate System
Curved coordinates are not automatically better.
Use the coordinate system that simplifies the geometry, field, or both.
A Reliable Coordinate-Selection Procedure
- Identify the geometry of the region or surface.
- Identify the symmetry of the field.
- Look for repeated combinations such as \(x^2+y^2\) or \(x^2+y^2+z^2\).
- Choose coordinates aligned with that symmetry.
- Write the correct differential element before integrating.
- Convert vector components if necessary.
- Check coordinate singularities and bounds.
A Reliable Formula-Selection Procedure
- Identify the coordinate system.
- Identify the field type: scalar or vector.
- Choose gradient, divergence, curl, or Laplacian as needed.
- Use the formula for that coordinate system.
- Do not substitute Cartesian formulas into cylindrical or spherical components.
- Simplify using symmetry whenever possible.
A Compact Coordinate Comparison
Cartesian:
\[ \boxed{ dV=dx\,dy\,dz }. \]Cylindrical:
\[ \boxed{ dV=r\,dr\,d\theta\,dz }. \]Spherical:
\[ \boxed{ dV=\rho^2\sin\phi\,d\rho\,d\theta\,d\phi }. \]A Compact Gradient Comparison
Cartesian:
\[ \nabla f = \mathbf{i}f_x + \mathbf{j}f_y + \mathbf{k}f_z. \]Cylindrical:
\[ \nabla f = \mathbf{e}_rf_r + \mathbf{e}_\theta\frac{1}{r}f_\theta + \mathbf{e}_zf_z. \]Spherical:
\[ \nabla f = \mathbf{e}_\rho f_\rho + \mathbf{e}_\theta \frac{1}{\rho\sin\phi}f_\theta + \mathbf{e}_\phi \frac{1}{\rho}f_\phi. \]Practice 1: Cylindrical Conversion
Convert
\[ (x,y,z) = (3,4,5) \]to cylindrical coordinates.
Practice 2: Spherical Radius
Find \(\rho\) for
\[ (x,y,z) = (2,-1,2). \]Practice 3: Cylindrical Volume Element
Write the differential volume element in cylindrical coordinates.
Practice 4: Spherical Surface Element
Write \(dS\) on the sphere
\[ \rho=R. \]Practice 5: Cylindrical Gradient
Let
\[ f(r,\theta,z)=r^2+z^2. \]Find \(\nabla f\).
Practice 6: Cylindrical Divergence
Let
\[ \mathbf{F} = r^2\mathbf{e}_r. \]Find \(\nabla\cdot\mathbf{F}\).
Practice 7: Spherical Divergence
Let
\[ \mathbf{F} = \rho^2\mathbf{e}_\rho. \]Find \(\nabla\cdot\mathbf{F}\).
Practice 8: Vector Conversion
Rewrite
\[ \mathbf{F} = \langle x,y,0\rangle \]in cylindrical coordinates.
Practice 9: Best Coordinate System
Which coordinate system is usually most natural for a field depending only on
\[ \sqrt{x^2+y^2+z^2}? \]Practice 10: Coordinate Singularity
Why is \(\theta\) undefined at
\[ r=0 \]in cylindrical coordinates?
Practice Answers
Practice 1
\[ r = \sqrt{3^2+4^2} = 5. \]Also,
\[ \theta = \arctan\left(\frac{4}{3}\right) \]in the first quadrant, and
\[ z=5. \]Therefore,
\[ \boxed{ (r,\theta,z) = \left( 5, \arctan\frac43, 5 \right) }. \]Practice 2
\[ \rho = \sqrt{2^2+(-1)^2+2^2} = \sqrt9 = \boxed{3}. \]Practice 3
\[ \boxed{ dV = r\,dr\,d\theta\,dz }. \]Practice 4
\[ \boxed{ dS = R^2\sin\phi \,d\theta\,d\phi }. \]Practice 5
We have
\[ f_r=2r, \qquad f_\theta=0, \qquad f_z=2z. \]Therefore,
\[ \boxed{ \nabla f = 2r\mathbf{e}_r + 2z\mathbf{e}_z }. \]Practice 6
Here
\[ F_r=r^2. \]Therefore,
\[ \nabla\cdot\mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} (r^3) = \frac{3r^2}{r} = \boxed{3r}. \]Practice 7
Here
\[ F_\rho=\rho^2. \]Therefore,
\[ \nabla\cdot\mathbf{F} = \frac{1}{\rho^2} \frac{\partial}{\partial\rho} (\rho^4) = \frac{4\rho^3}{\rho^2} = \boxed{4\rho}. \]Practice 8
\[ \boxed{ \mathbf{F} = r\mathbf{e}_r }. \]Practice 9
The natural choice is
\[ \boxed{ \text{spherical coordinates} }. \]Practice 10
At the \(z\)-axis, there is no unique direction from the axis toward the point in the \(xy\)-plane.
Therefore the azimuthal angle \(\theta\) is not uniquely determined.
What to Remember
Cartesian coordinates use fixed basis vectors and unit scale factors.
Cylindrical coordinates use
\[ \boxed{ h_r=1, \qquad h_\theta=r, \qquad h_z=1 }. \]Spherical coordinates use
\[ \boxed{ h_\rho=1, \qquad h_\theta=\rho\sin\phi, \qquad h_\phi=\rho }. \]- Coordinate systems describe the same geometry in different ways.
- Curved coordinates introduce position-dependent basis vectors and scale factors.
- Cylindrical coordinates fit axial symmetry.
- Spherical coordinates fit radial symmetry.
- Jacobians come from coordinate scale factors.
- Gradient, divergence, curl, and Laplacian formulas change with coordinates.
- Flux and surface elements simplify when normals align with coordinate basis directions.
- Coordinate singularities are not automatically physical singularities.
- The best coordinate system is the one that simplifies the geometry and field together.
Where We Go Next
Coordinate systems provide the language needed to express vector-calculus operators efficiently in geometries that match real physical problems.
The next lesson, Physical Field Models, will combine scalar and vector fields, gradients, divergence, curl, line integrals, surface integrals, flux, coordinate systems, and the major integral theorems into concrete models from mechanics, fluids, gravitation, electromagnetism, heat, and transport.