Introduction to Electrodynamics · Problem 1.48

Vector Analysis — David J. Griffiths

Physics · dirac-delta-function-3d

Problem

Evaluate the following integrals: (a)

\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,

a

is its magnitude, and the integral is over all space. (b)

\int_{\mathcal{V}} |\mathbf{r} - \mathbf{b}|^2 \delta^3(5\mathbf{r})\,d\tau

, where

\mathcal{V}

is a cube of side 2, centered on the origin, and

\mathbf{b} = 4\,\hat{\mathbf{y}} + 3\,\hat{\mathbf{z}}

. (c)

\int_{\mathcal{V}} \left[r^4 + r^2(\mathbf{r} \cdot \mathbf{c}) + c^4\right] \delta^3(\mathbf{r} - \mathbf{c})\,d\tau

, where

\mathcal{V}

is a sphere of radius 6 about the origin,

\mathbf{c} = 5\,\hat{\mathbf{x}} + 3\,\hat{\mathbf{y}} + 2\,\hat{\mathbf{z}}

, and

c

is its magnitude. (d)

\int_{\mathcal{V}} \mathbf{r} \cdot (\mathbf{d} - \mathbf{r})\,\delta^3(\mathbf{e} - \mathbf{r})\,d\tau

, where

\mathbf{d} = (1, 2, 3)

\mathbf{e} = (3, 2, 1)

, and

\mathcal{V}

is a sphere of radius 1.5 centered at

(2, 2, 2)