Mathematical Methods for Physics and Engineering

1. Some integrals:

(1) 2-D integrals:

$\iint_{D} f(x,y) \, dx \, dy$;

(2) 3-D integrals:

$\iiint_{V} f(x,y,z) \, dx \, dy \, dz$;

(3) Area integrals:

$\iint_{D} g(x,y) \, dA$;

(4) Volume integrals (2-D):

$\iint_{D} h(x,y) \, dA$;

(5) Volume integrals (3-D):

$\iiint_{V} h(x,y,z) \, dV$;

(6) Center of mass:

$\bar{x}=\frac{1}{M}\iint_{D} x\rho(x,y)\,dA$, $\bar{y}=\frac{1}{M}\iint_{D} y\rho(x,y)\,dA$;

(7) Mean values of functions:

$\langle f\rangle = \frac{1}{A}\iint_{D} f(x,y)\,dA$ or $\langle f\rangle = \frac{1}{V}\iiint_{V} f(x,y,z)\,dV$.

A good example may be mean values of density: $\bar{\rho} = \frac{1}{V} \iiint_{V} \rho(x,y,z)\,dV$.

Notes:

a. Be careful about substituting some equations among provided conditions in some problems. For example, substituting $y^2 = 4ax$ into $x + z = a$, then the new surface must still encompass the volume. If the surface is not arranged correctly then the calculation will be wrong.

b. The correct order of integration is important; sometimes it saves time.

c. The selection of (4) and (5) is important; this also saves your time.

2. Variables change:

(1) 2-D:

if $x = x(u,v)$ and $y = y(u,v)$, then

$dx\,dy = \left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv$;

Example: $\iint_{D} f(x,y)\,dx\,dy = \iint_{S} f(x(u,v),y(u,v))\left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv$.

(2) 3-D:

if $x = x(u,v,w)$, $y = y(u,v,w)$ and $z = z(u,v,w)$, then

$dx\,dy\,dz = \left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right| du\,dv\,dw$.

Example: find the moment of inertia of a sphere with radius $a$ and mass $M$. For a solid sphere about its diameter,

$I = \tfrac{2}{5}Ma^{2}$.

(3) Properties of Jacobians:

$\frac{\partial(x,y)}{\partial(u,v)} \frac{\partial(u,v)}{\partial(x,y)} = 1$, and

$\frac{\partial(x,y)}{\partial(u,v)} = \left(\frac{\partial(u,v)}{\partial(x,y)}\right)^{-1}$.