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Mathematical Methods for Physics and Engineering 1. Basic definitions: (0) Definition of linear vector space: a set of $\mathbf{a}$,$\mathbf{b}$,$\mathbf{c}$,$\dots$ forms a linear vector space providing that: (1) Linearly dependent: $$\alpha\mathbf{a}+\beta\mathbf{b}+\cdots+\sigma\mathbf{s}=\mathbf{0}$$ (2) A complete set: $$\alpha\mathbf{e}_1+\beta\mathbf{e}_2+\cdots+\sigma\mathbf{e}_N+\chi\mathbf{x}=\mathbf{0}$$ or $$\mathbf{x}=x_{1}\mathbf{e}_{1}+x_{2}\mathbf{e}_{2}+\cdots+x_{N}\mathbf{e}_{N}=\sum_{i=1}^{N}x_{i}\mathbf{e}_{i}$$where $\alpha, \beta,\ldots,\chi$ are…

Mathematical Methods for Physics and Engineering 1. Some conclusions: (1) Ratio theorem: Line segment AB is divided into AP and PB. AP:PB = $\lambda:\mu$. Then $\mathbf{OP}=\frac{\mu}{\lambda+\mu} \mathbf{OA}+\frac{\lambda}{\lambda+\mu} \mathbf{OB}$. (2) Centroid G of a triangle: $$\mathbf{g}=\frac{1}{3}(\mathbf{a}+\mathbf{b}+\mathbf{c})$$. (3) The cosine of the…

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):…

Mathematical Methods for Physics and Engineering 1. Some properties of partial differentiation: 1.1 Clairaut’s theorem: Let $f(x,y)$ be a function. Suppose $f$ has continuous second-order partial derivatives in a neighborhood of a point $(x_0,y_0)$. Then: $$\frac{\partial^2 f}{\partial x\partial y} =…

Mathematical Methods for Physics and Engineering 1. Basic sum of series [Page 117-118]: Basic series: $$\begin{array}{ll} Arithmetic\ series:S_N=\frac{N}{2}(first\ term+last\ term)\\ Geometric\ series:S_N=\frac{a(1-r^N)}{1-r} \end{array}$$ Especially, $S_N=\frac{a}{1-r}$ for infinite geometric series with $|r|<1$. Arithmetico-geometric series: $$\begin{array}{ll} S_N & =\sum\limits_{n=0}^{N-1}(a+nd)r^n\\ & =\mathbf{\frac{a-[a+(N-1)d]r^N}{1-r}+\frac{rd(1-r^{N-1})}{(1-r)^2}} \end{array}$$…

Mathematical Methods for Physics and Engineering 1. Complex number division [Page 90-91]: $\frac{z_1}{z_2}=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}$ Especially for conjugate division: $\frac{z}{z^*}=\frac{x^2-y^2}{x^2+y^2}+i\frac{2xy}{x^2+y^2}$ 2. Complex number logarithms [Page 100]: $z=re^{i(\theta+2n\pi)},\ Lnz=lnr+i(\theta+2n\pi)$ 3. Complex number applied to differentiation and integration [Page 101]: Sometimes complementing trigonometric functions…