Mathematical Methods for Physics and Engineering
1. Some integrals:
(1) 2-D integrals: I=\int_{x=a}^{x=b}\left\{\int_{y=y_1(x)}^{y=y_2(x)}f(x,y) dy\right\} dx;
(2) 3-D integrals: I=\int_{x_1}^{x_2}dx\int_{y_1(x)}^{y_2(x)}dy\int_{z_1(x,y)}^{z_2(x,y)}dz f(x,y,z);
(3) Area integrals: A=\int_RdA=\int\int_Rdx dy;
(4) Volume integrals(2-D): V=\int_Rz dA=\int\int_Rf(x,y) dx dy;
(5) Volume integrals(3-D): V=\int_RdV=\int\int\int_Rdx dy dz;
(6) Center of mass: \begin{cases} \bar{x} \int d M & =\int x d M \\ \bar{y} \int d M & =\int y d M \\ \bar{z} \int d M & =\int z d M \end{cases} \Rightarrow \overline{\mathbf{r}}=\frac{1}{M} \int \mathbf{r} d M ;
(7) Mean values of functions: \bar{f} \int_R d A=\int_R f(x, y) d A or \bar{f} \int_R d V=\int_R f(x, y, z) d V.
A good example may be mean values of density: \bar{\rho} \int_R d V=\int_R \rho(x, y, z) d V.
Notes:
a. Be careful about substituting some equations among provided conditions in some problems. For example, substituting y2 = 4ax into x + z = a, then the new surface must still encompass the volume. If the surface is not arranged correctly then calculation will be wrong.
b. The correct order of integral is important, sometimes saving your time.
c. The selection of (4) and (5) is important, this also saves your time.
2. Variables change:
(1) 2-D: if \begin{cases} u =u(x,y) \\ v =v(x,y) \end{cases} , then I=\iint_R f(x, y) d x d y=\iint_{R^{\prime}} g(u, v)\left|\frac{\partial(x, y)}{\partial(u, v)}\right| d u d v where \begin{cases} \frac{\partial(x, y)}{\partial(u, v)}=\left|\begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{array}\right|\\ f(x,y)=g(u,v) \end{cases};
Example: I=\int_{-\infty}^{\infty} e^{-x^2} d x.
(2) 3-D: if \begin{cases} u & =u(x,y,z) \\ v & =v(x,y,z) \\ w & =w(x,y,z) \end{cases} , then I=\iint_V f(x, y,z) d x d y d z =\iint_{V^{\prime}} g(u, v, w)\left|\frac{\partial(x, y, z)}{\partial(u, v, w)}\right|d u d v d w where \begin{cases} \frac{\partial(x, y, z)}{\partial(u, v, w)} =\left|\begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \\ \frac{\partial x}{\partial w} & \frac{\partial y}{\partial w} & \frac{\partial z}{\partial w} \end{array}\right|\\ f(x,y,z)=g(u,v,w) \end{cases} .
Example: find the moment of inertia of a sphere with radius a and mass M.
(3) Properties of Jacobians: \begin{cases} J_{xy}J_{yx}=J_{xx}=1\\ J_{xy}J_{yz}=J_{xz} \end{cases}.
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