Mathematics for Physics I - Vector Calculus

Vector Fields

Vector calculus deals with differentiation and integration of vector fields, which are functions that assign a vector to each point in space.

Definition

A vector field in 2D is a function \(\vec{F}(x, y) = P(x, y)\hat{i} + Q(x, y)\hat{j}\).

A vector field in 3D is a function \(\vec{F}(x, y, z) = P(x, y, z)\hat{i} + Q(x, y, z)\hat{j} + R(x, y, z)\hat{k}\).

Examples in physics include gravitational fields, electric fields, magnetic fields, and fluid velocity fields.

Visualization

Vector fields are visualized by drawing arrows at various points in space, where the length and direction of the arrow represent the magnitude and direction of the vector at that point.

This provides a visual representation of the field and its properties, such as sources, sinks, and circulation.

Interactive visualization of a 2D vector field

Gradient, Divergence, and Curl

These are three fundamental vector differential operators that describe the behavior of scalar and vector fields.

Gradient (∇f)

The gradient of a scalar field f(x, y, z) is a vector field:

\[ \nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k} \]

The gradient points in the direction of the greatest rate of increase of f, and its magnitude is that rate of increase.

It is perpendicular to the level surfaces of f.

Divergence (∇ ⋅ F)

The divergence of a vector field \(\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}\) is a scalar field:

\[ \nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]

Divergence measures the net outward flux per unit volume at a point.

If \(\nabla \cdot \vec{F} > 0\), the point is a source.

If \(\nabla \cdot \vec{F} < 0\), the point is a sink.

If \(\nabla \cdot \vec{F} = 0\), the field is solenoidal (incompressible).

Curl (∇ × F)

The curl of a vector field \(\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}\) is a vector field:

\[ \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} \] \[ = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\hat{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\hat{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\hat{k} \]

Curl measures the circulation or rotation of the vector field at a point.

If \(\nabla \times \vec{F} = \vec{0}\), the field is irrotational (conservative).

Important Identities

\[ \nabla \cdot (\nabla \times \vec{F}) = 0 \quad \text{(divergence of curl is zero)} \] \[ \nabla \times (\nabla f) = \vec{0} \quad \text{(curl of gradient is zero)} \]

These identities are crucial in vector calculus and have important physical implications, such as the non-existence of magnetic monopoles (\(\nabla \cdot \vec{B} = 0\)) and the conservative nature of electrostatic fields (\(\nabla \times \vec{E} = \vec{0}\) in static cases).

Line Integrals

Line integrals are used to integrate a function along a curve in space.

Line Integral of a Scalar Field

If C is a curve parameterized by \(\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}\) for a ≤ t ≤ b, then:

\[ \int_C f(x, y, z) \, ds = \int_a^b f(x(t), y(t), z(t)) \|\vec{r}\'(t)\| \, dt \]

Where \(ds = \|\vec{r}\'(t)\| \, dt = \sqrt{(x\'(t))^2 + (y\'(t))^2 + (z\'(t))^2} \, dt\) is the arc length element.

This can represent the mass of a wire with variable density, or the average value of a function along a curve.

Line Integral of a Vector Field

If \(\vec{F}\) is a vector field and C is a curve parameterized by \(\vec{r}(t)\), then:

\[ \int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}\'(t) \, dt \]

This represents the work done by the force field \(\vec{F}\) in moving a particle along the curve C.

If \(\vec{F} = \nabla f\) (conservative field), then:

\[ \int_C \vec{F} \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)) \]

This is the Fundamental Theorem for Line Integrals, showing that the line integral of a conservative field is path-independent.

Visualization of a line integral of a vector field along a curve

Surface Integrals

Surface integrals are used to integrate a function over a surface in space.

Surface Integral of a Scalar Field

If S is a surface parameterized by \(\vec{r}(u, v) = x(u, v)\hat{i} + y(u, v)\hat{j} + z(u, v)\hat{k}\), then:

\[ \iint_S f(x, y, z) \, dS = \iint_D f(\vec{r}(u, v)) \|\vec{r}_u \times \vec{r}_v\| \, du \, dv \]

Where \(dS = \|\vec{r}_u \times \vec{r}_v\| \, du \, dv\) is the surface area element.

This can represent the mass of a thin shell with variable density, or the average value of a function over a surface.

Surface Integral of a Vector Field (Flux)

If \(\vec{F}\) is a vector field and S is an oriented surface with unit normal vector \(\hat{n}\), then the flux of \(\vec{F}\) across S is:

\[ \iint_S \vec{F} \cdot d\vec{S} = \iint_S \vec{F} \cdot \hat{n} \, dS = \iint_D \vec{F}(\vec{r}(u, v)) \cdot (\vec{r}_u \times \vec{r}_v) \, du \, dv \]

This represents the net flow of the vector field \(\vec{F}\) across the surface S.

For example, the flux of an electric field across a closed surface is related to the enclosed charge by Gauss's Law.

Integral Theorems

These theorems relate integrals over different dimensions and are fundamental in vector calculus.

Green's Theorem

Relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C:

\[ \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \]

This is a 2D version of Stokes' Theorem.

Stokes' Theorem

Relates the surface integral of the curl of a vector field \(\vec{F}\) over a surface S to the line integral of \(\vec{F}\) around the boundary curve C of S:

\[ \iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r} \]

This theorem is crucial in electromagnetism, relating the circulation of the magnetic field to the current density (Ampère's Law).

Divergence Theorem (Gauss's Theorem)

Relates the flux of a vector field \(\vec{F}\) through a closed surface S to the triple integral of the divergence of \(\vec{F}\) over the volume V enclosed by S:

\[ \oiint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) \, dV \]

This theorem is fundamental in fluid dynamics and electromagnetism (Gauss's Law for electric and magnetic fields).

Applications in Physics

Electromagnetism (Maxwell's Equations)

Maxwell's equations, the foundation of classical electromagnetism, are expressed using vector calculus:

\[ \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \quad \text{(Gauss's Law)} \] \[ \nabla \cdot \vec{B} = 0 \quad \text{(Gauss's Law for magnetism)} \] \[ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \quad \text{(Faraday's Law)} \] \[ \nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t} \quad \text{(Ampère-Maxwell Law)} \]

These equations describe how electric and magnetic fields are generated and interact with charges and currents.

Fluid Dynamics

The continuity equation for fluid flow is:

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \]

Where ρ is the fluid density and \(\vec{v}\) is the velocity field.

For incompressible flow (ρ = constant), this simplifies to \(\nabla \cdot \vec{v} = 0\).

The Navier-Stokes equations, which describe fluid motion, also heavily rely on vector calculus.

Heat Transfer

Fourier's Law of heat conduction states that the heat flux \(\vec{q}\) is proportional to the negative gradient of temperature T:

\[ \vec{q} = -k \nabla T \]

Where k is the thermal conductivity.

The heat equation, which describes how temperature changes over time, is:

\[ \frac{\partial T}{\partial t} = \alpha \nabla^2 T \]

Where α is the thermal diffusivity and \(\nabla^2 = \nabla \cdot \nabla\) is the Laplacian operator.

Quantum Mechanics

The Schrödinger equation, which describes the evolution of quantum systems, involves the Laplacian operator:

\[ i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V\Psi \]

Where Ψ is the wave function, ħ is the reduced Planck constant, m is the mass, and V is the potential energy.

The probability current density is also defined using vector calculus concepts.

Key Insight:

Vector calculus provides the mathematical framework for describing and analyzing physical phenomena that involve fields and their interactions. From the fundamental forces of nature to the behavior of fluids and heat, vector calculus allows us to formulate concise and powerful laws that govern the physical world. The concepts of gradient, divergence, curl, and the integral theorems are indispensable tools for physicists and engineers.

Practice Problems

Test your understanding of vector calculus with these practice problems:

Find the gradient of the scalar field f(x, y, z) = x²y + yz³ + zx.
Calculate the divergence and curl of the vector field \(\vec{F}(x, y, z) = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}\).
Evaluate the line integral \(\int_C \vec{F} \cdot d\vec{r}\) where \(\vec{F}(x, y) = (x+y)\hat{i} + (y-x)\hat{j}\) and C is the line segment from (0,0) to (1,1).
Use Green's Theorem to evaluate \(\oint_C (x^2y \, dx + xy^2 \, dy)\) where C is the boundary of the square with vertices (0,0), (1,0), (1,1), and (0,1).
Use the Divergence Theorem to calculate the flux of \(\vec{F}(x, y, z) = x\hat{i} + y\hat{j} + z\hat{k}\) across the surface of the unit sphere.