Denotation: $J = {δ(u,v)}/{δ(x,y)} = |\table u_x,u_y;v_x,v_y|$ Chain rule of jacobians: x,y -> u,v -> r,s ${δ(x,y)}/{δ(u,v)} . {δ(u,v)}/{δ(r,s)} = {δ(x,y)}/{δ(r,s)}$ J. J’ = 1 Jacobian of implicit functions: $u_1, u_2, u_3$ be implicit functions of variables $x_1, x_2, x_3$ connected by $f_1, f_2, f_3$ such that $f_1(u_1,u_2,u_3,x_1,x_2,x_3)=0$ $f_2(u_1,u_2,u_3,x_1,x_2,x_3)=0$ $f_3(u_1,u_2,u_3,x_1,x_2,x_3)=0$ ${δ(u_1,u_2,u_3)}/{δ(x_1,x_2,x_3)} = (-1)^3 {{δ(f_1,f_2,f_3)}/{δ(x_1,x_2,x_3)}}/{{δ(f_1,f_2,f_3)}/{δ(u_1,u_2,u_3)}}$ General: ${δ(u_1,u_2,u_3,…,u_n)}/{δ(x_1,x_2,x_3,…,x_n)} = (-1)^n {{δ(f_1,f_2,f_3,…,f_n)}/{δ(x_1,x_2,x_3,…,x_n)}}/{{δ(f_1,f_2,f_3,…,f_n)}/{δ(u_1,u_2,u_3,…,u_n)}}$ Four variables u,v,x,y related by implicit functions: $f_1(u,v,x,y)=0$ & $f_2(u,v,x,y) = 0$ ${δu}/{δx} = – {{δ(f_1,f_2)}/{δ(x,y)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$ ${δv}/{δx} = – {{δ(f_1,f_2)}/{δ(u,x)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$ ${δu}/{δy} = – {{δ(f_1,f_2)}/{δ(y,v)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$ ${δv}/{δy} = – {{δ(f_1,f_2)}/{δ(u,y)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$ Functional Dependence $J = {δ(f_1,f_2)}/{δ(x,y)} = {δ(u,v)}/{δ(x,y)} = 0$ ${δ(f_1,f_2)}/{δ(x,y)} = 0, {δ(f_1,f_2)}/{δ(y,z)} ...
Read more
Examples for Rules of Partial Differentiation: Derivative of Sum: $δ/{δx} (u±v) = {δu}/{δx}±{δv}/{δx} $ Derivative of Product: $δ/{δx} (uv) = u {δv}/{δx} + v {δu}/{δx}$ Derivative of Quotient: $δ/{δx} (u/v) = {v {δu}/{δx} – u {δv}/{δx}}/{v^2}$ If k is constant, then $δ/{δx} (ku) = k {δu}/{δx} $ Derivative of constant is 0 $δ/{δx} [f(x,y,z)]^n = n[f(x,y,z)]^{n-1} {δf}/{δx}$ $δ/{δx} [1/{f(x,y,z)}] = {-1}/{[f(x,y,z)]^2} {δf}/{δx}$ $δ/{δx} [√{f(x,y,z)}] = 1/{2√{f(x,y,z)} {δf}/{δx}}$ $ δ/{δx} [log f(x,y,z)] = 1/{f(x,y,z)} {δf}/{δx} $ $ δ/{δx} [a^{f(x,y,z)}] = a^{f(x,y,z)} log a {δf}/{δx} $ $ δ/{δx} [sin f(x,y,z)] = cos f(x,y,z) {δf}/{δx} $ $ δ/{δx} [cos f(x,y,z)] = -sin f(x,y,z) ...
Read more
Seven indeterminate forms: $0/0, ∞/∞, 0*∞, ∞ – ∞, 0^0, ∞^0, 1^∞$ 0/0 Form: L’Hospital Rule $\lim↙{x→a} {f(x)}/{g(x)} = \lim↙{x→a} {f'(x)}/{g'(x)}$ ∞/∞ Form: Write: $ {f(x)}/{g(x)} \text" as " {{1}/{g(x)}}/{{1}/{f(x)}}$ 0 x ∞ Form: f(x) . g(x) = ${f(x)}/{{1}/{g(x)}}$ ∞ – ∞ Form: Convert into any of the first 2 forms n apply L’Hospital Rule $0^0$ Form: $L = e^b$ where, $b = \lim↙{x→a} g(x) log f(x)$
Maclaurin’s Theorem: $f(x) = f(0) + xf'(0) + x^2/{2!} f”(0) + x^3/{3!} f”'(0) +…+ x^n/{n!} f^n(0) +…$ Standard Expansions: $sinx = x – x^3/{3!} + x^5/{5!} – x^7/{7!} +…$ $cosx = 1 – x^2/{2!} + x^4/{4!} – x^6/{6!} + …$ $tanx = x + x^3/3 + 2/{15} x^5 + {17}/{315} x^7 + …$ $e^x = 1 + x +x^2/{2!} + x^3/{3!} + x^4/{4!} + …$ $ sinhx = x + x^3/{3!} + x^5/{5!} + x^7/{7!} +… $ $coshx = 1 + x^2/{2!} + x^4/{4!} + x^6/{6!} + …$ $tanhx = x – x^3/3 + 2/{15} x^5 – {17}/{315} x^7 + ...
Read more
$$ {d^n}/{dx^n} [a^x] = a^x (log_{e}a)^n $$ $$ {d^n}/{dx^n} [(ax + b)^m] = m(m-1)(m-2)…(m-n+1)a^n (ax+b)^{m-n} $$ $$ {d^n}/{dx^n} [{1}/{(ax+b)^m}] = {(-1)^n m(m+1)(m+2)…(m+n-1)a^n}/{(ax+b)^{m+n}} $$ Case: m > n $$ {d^n}/{dx^n} [(ax+b)^m] = {m! a^{n} (ax+b)^{m-n}}/{(m-n)!} $$ Case: m = n $$ {d^n}/{dx^n} [(ax+b)^n] = n! a^n$$ if a=1 and b = 0 $$ {d^n}/{dx^n} [x^n] = n! $$ Case: m is positive and m
Infinite series is denoted as: $$∑↙{n=1}↖∞ u_n$$ or $$∑u_n$$ Sum of first n terms in series is denoted by: $S_n = u_1 + u_2 + … + u_n$ Convergent series: $S_n$ tends to finite limit S as n tends to infinity. $S = \lim↙{n→∞} S_n$ Divergent series: $S_n$ tends to +∞ or -∞ as n tends to infinity. $\lim↙{n→∞} S_n = +∞ or -∞ $ Oscillatory Series: $S_n$ neither tends to a finite limit nor to +∞ or -∞ as n tends to infinity. General: * $\lim S_n$ = finite quantity -> series is convergent. * Series is positive and ...
Read more
cosx = ${e^{ix}+e^{-ix}}/{2}$ sinx = ${e^{ix}-e^{-ix}}/{2i}$ tanx = ${e^{ix}-e^{-ix}}/{i(e^{ix}+e^{-ix})}$ sinhx = ${e^{x}-e^{-x}}/{2}$ coshx = ${e^{x}+e^{-x}}/{2}$ tanhx = ${e^{x}-e^{-x}}/{e^{x}+e^{-x}}$ Formulae of Hyperbolic Fucntions: $cosh^{2}x – sinh^{2}x$ = 1 $sech^{2}x$ = 1 – $tanh^{2}x$ $coth^{2}x$ = 1 + $cosech^{2}x$ sinh(x±y) = sinhx coshy ± coshx sinhy cosh(x±y) = coshx coshy ± sinhx sinhy $\table cosh2x,=, cosh^{2}x + sinh^{2}x; \ ,=,2cosh^{2}x-1;,=,1+2sinh^{2}x$ sinh2x = 2 sinhx coshx tanh(x±y) = ${tanhx ± tanhy}/{1± tanhx tanhy}$ sinhx = ${2tanhx/2}/{1-tanh^{2}x/2}$ coshx = ${1+tanh^{2}x/2}/{1-tanh^{2}x/2}$ tanhx = ${2tanhx/2}/{1 + tanh^{2}x/2}$ sinh3x = 3sinhx + 4$sinh^{3}x$ cosh3x = 4$cosh^{3}x$ – 3 coshx Separation of real & imaginary formulae sin(x+iy) = ...
Read more
z = x + iy with (x,y) as point coordinates. z = r(cosθ + isinθ), where x = r cosθ & y = r sinθ $r = √{x^2+y^2}$ θ = $tan^{-1}(y/x)$ amp(z) / arg(z) = θ As arg(z) is multivalued function & cos, sin have 2π radians period: arg(z) = θ+2nπ, n = 0,1,2,… θ+2nπ is general value and value of θ in between -π & π is principle value Principle value of arguement (1)$I^{st}$ quadrant (x>0,y>0): θ = arg(z) = $tan^{-1}(y/x)$ (2)$II^{nd}$ quadrant (x0): θ = arg(z) = π – $tan^{-1}(y/x)$ (3)$III^{rd}$ quadrant (x
Note: $ λ_1 $ x $λ_2$ x…x $λ_n$ = |A| If $ λ_1, λ_2, …, λ_n $ are eigen values of A, then $ 1/λ_1, 1/λ_2, …, 1/λ_n $ are eigen values of $A^{-1}$ Eigen values of A and A’ are same. $A^{-1}$ exists iff $λ_j$ ≠ 0, j = 1,2,.., n. Orthogonal eigen vectors: Two eigen vectors $X_1$ & $X_2$ are said to be orthogonal if $X’_1 X_2 = 0$ Note: Eigen values may be zero; an eigen vector may not be the zero vector. For 2 x 2 matrix, $λ^2 – S_1 λ + |A| = 0$ $S_1$ ...
Read more
Condition for consistency of non-homogenous equation Consider, AX = B m = total no. of equations, n = toal no. of unknowns Case: m ≠ n ρ(A) = ρ(A, B) i.e. system is consistent. ρ(A) ≠ ρ(A, B) i.e. system is inconsistent and hence no solution ρ(A) = ρ(A, B) = n = total no. of unknowns, hence unique solution i.e. only one solution ρ(A) = ρ(A, B) = r < n, hence infinite solutions which can be represented parametrically by employing some parameter t. Thus, r unknowns can be expressed in terms of remaining (n-r) unknowns. Case: m = ...
Read more