Dirac Delta Function

Exercise 1 : Basic Delta Integrals

Evaluate the following integrals:

  1. $\displaystyle \int_{2}^{6} (3x^2 - 2x - 1) , \delta(x - 3) , dx$
  2. $\displaystyle \int_{0}^{5} \cos x ; \delta(x - \pi) , dx$
  3. $\displaystyle \int_{-1}^{3} x^3 , \delta(x + 1) , dx$
  4. $\displaystyle \int_{-\infty}^{\infty} \ln(x + 3) , \delta(x + 2) , dx$
  5. $\displaystyle \int_{0}^{\infty} e^{-x} , \delta(x - 2) , dx$

Exercise 2 : Delta with Scaled & Shifted Arguments

Use $\delta(ax) = \frac{1}{|a|}\delta(x)$ and similar properties to evaluate:

  1. $\displaystyle \int_{-2}^{2} (2x + 3) , \delta(3x) , dx$
  2. $\displaystyle \int_{0}^{2} (x^3 + 3x + 2) , \delta(1 - x) , dx$
  3. $\displaystyle \int_{-1}^{1} 9x^2 , \delta(3x + 1) , dx$
  4. $\displaystyle \int_{-\infty}^{\infty} f(x) , \delta(ax - b) , dx$ for $a \ne 0$ (general formula)
  5. $\displaystyle \int_{0}^{4} \sin(\pi x) , \delta!\left(x - \frac{1}{2}\right) dx$

Exercise 3 : Fundamental Properties

Let $\delta(x - b)$ be a Dirac delta centered at $x = b$.

  1. $\displaystyle \int_{-\infty}^{\infty} \delta(x - b) , dx = ;?$
  2. If $f(x)$ is continuous at $x = b$, justify the sampling property: $\displaystyle \int f(x) \delta(x-b),dx = f(b)$ in one sentence.
  3. Prove: $\displaystyle x \frac{d\delta(x)}{dx} = -\delta(x)$
    Hint: Use integration by parts with test functions.

Exercise 4 : Heaviside Step Function Relation

The Heaviside step function is:

$$ \theta(x) = \begin{cases} 1, & x > 0, \\ 0, & x \leq 0. \end{cases} $$

Show that $\displaystyle \frac{d\theta}{dx} = \delta(x)$, arguing from the fundamental theorem of calculus and the sampling property.

Exercise 5 : General Composition Rule

Let $f(x)$ be smooth with a simple zero at $x_0$, so $f(x_0) = 0$ but $f’(x_0) \neq 0$. Show:

$$ \delta\big(f(x)\big) = \frac{\delta(x - x_0)}{|f'(x_0)|}. $$

Use this to re-derive the scaling rule $\delta(ax) = \frac{1}{|a|}\delta(x)$.

Exercise 6 : 3D Delta Integrals

Evaluate the following 3D integrals over all space, unless specified otherwise:

  1. $\displaystyle \int (r^2 + \mathbf{r}\cdot\mathbf{a} + a^2), \delta^3(\mathbf{r} - \mathbf{a}) , d\tau$,
    where $\mathbf{a}$ is a fixed vector and $a = |\mathbf{a}|$.
  2. $\displaystyle \int_V |\mathbf{r} - \mathbf{b}|^2 , \delta^3(\mathbf{r}) , d\tau$,
    with $V$ a cube of side 2 centered on the origin, $\mathbf{b} = 4\hat{y} + 3\hat{z}$.
  3. $\displaystyle \int_V \big[r^4 + r^2 (\mathbf{r}\cdot\mathbf{c}) + c^4\big] \delta^3(\mathbf{r} - \mathbf{c}) , d\tau$,
    where $V$ is a sphere radius 6 centered at origin, $\mathbf{c} = 5\hat{x}+3\hat{y}+2\hat{z}$, $c = |\mathbf{c}|$.
  4. $\displaystyle \int_V (\mathbf{r}\cdot\mathbf{d}), \delta^3(\mathbf{e} - \mathbf{r}) , d\tau$,
    with $\mathbf{d} = (1,2,3)$, $\mathbf{e} = (3,2,1)$, $V$ a sphere radius 1.5 centered at $(2,2,2)$.

Exercise 7 : Charge Density Expressions

Write an expression for the volume charge density $\rho(\mathbf{r})$ for:

  1. A point charge $q$ at $\mathbf{r}’$.
  2. An electric dipole: charge $-q$ at origin and $+q$ at $\mathbf{a}$.
  3. A uniform spherical shell radius $R$, total charge $Q$, centered at origin (in spherical coordinates).
  4. A uniform infinite line charge along the $z$-axis with linear density $\lambda$.

Ensure in each case that $\int \rho , d\tau$ gives the correct total charge.

Exercise 8 : Delta Identity from Divergence Theorem

Let $\mathbf{v} = \frac{\hat{\mathbf{r}}}{r^2}$.

  1. Compute $\nabla \cdot \mathbf{v}$ for $r \neq 0$.
  2. By the divergence theorem, show that $\int_{\mathcal{V}} \nabla \cdot \mathbf{v} , d\tau = 4\pi$ if $\mathcal{V}$ contains the origin, and 0 otherwise.
  3. Conclude that $\nabla \cdot \left( \frac{\hat{\mathbf{r}}}{r^2} \right) = 4\pi \delta^3(\mathbf{r})$.

Exercise 9 : Application of Delta Divergence Identity

Evaluate:

$$ J = \int_{V} e^{-r} \left[ \nabla \cdot \left( \frac{\hat{\mathbf{r}}}{r^2} \right) \right] d\tau $$

where $V$ is a sphere of radius $R$ centered at the origin, using:

  1. The identity: $\nabla \cdot \left( \frac{\hat{\mathbf{r}}}{r^2} \right) = 4\pi \delta^3(\mathbf{r})$
  2. Direct integration with the divergence theorem, treating the origin carefully.

Exercise 10 : Delta in Different Coordinate Systems

The 3D delta in Cartesian, spherical, and cylindrical coordinates:

Show that $\delta^3(\mathbf{r} - \mathbf{r}’) = \frac{\delta(r-r’),\delta(\theta-\theta’),\delta(\phi-\phi’)}{r^2 \sin\theta}$ in spherical coordinates for $\mathbf{r}’ \neq 0$, and explain the geometric origin of the factor $1/(r^2\sin\theta)$.

Exercise 11 : Delta as Limit of Functions

Verify that

$$ \lim_{\epsilon \to 0} \frac{1}{\sqrt{2\pi\epsilon}} e^{-x^2/(2\epsilon)} $$

acts as a 1D Dirac delta by checking:

  1. Integral over $\mathbb{R}$ is 1 for any $\epsilon>0$,
  2. For smooth $f(x)$, $\displaystyle \lim_{\epsilon \to 0} \int f(x) g_\epsilon(x),dx = f(0)$.

Exercise 12 : Fourier Representation of Delta

In quantum mechanics, the identity $\int_{-\infty}^\infty e^{ik(x-x’)} dk = 2\pi, \delta(x-x’)$ is used frequently.

  1. Relate this to the Fourier representation of the delta function.
  2. Use it to evaluate $\int_{-\infty}^\infty \cos(kx),dk$ as a distribution.

Exercise 13 : Advanced Delta Manipulations

  1. Compute $\displaystyle \int_{-\infty}^{\infty} \delta’(x) f(x) , dx$ in terms of $f’(0)$.
  2. Verify: $\displaystyle \delta(x^2 - a^2) = \frac{1}{2|a|} \bigl[ \delta(x - a) + \delta(x + a) \bigr]$.
  3. Evaluate: $\displaystyle \int_{-\pi}^{\pi} \sin^2 x ; \delta’!(x) , dx$.
  4. Show: $\displaystyle \frac{d}{dx} \theta(x^2 - 1) = \delta(x - 1) - \delta(x + 1)$.