Jacobians, Errors and Approximations, Maxima and Minima
December 8, 2016Denotation: $J = {δ(u,v)}/{δ(x,y)} = |\table u_x,u_y;v_x,v_y|$ Chain rule of jacobians: x,y -> u,v -> r,s ${δ(x,y)}/{δ(u,v)} . {δ(u,v)}/{δ(r,s)} =... View Article
Categories for 1st Year Maths 1
Partial Differentiation and Applications
December 8, 2016Examples for Rules of Partial Differentiation: Derivative of Sum: $δ/{δx} (u±v) = {δu}/{δx}±{δv}/{δx} $ Derivative of Product: $δ/{δx} (uv) =... View Article
Indeterminate Forms
December 8, 2016Seven 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}... View Article
Taylor and Maclaurin Theorem
December 8, 2016Maclaurin’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... View Article
Successive Differentiation
December 8, 2016$$ {d^n}/{dx^n} [a^x] = a^x (log_{e}a)^n $$ $$ {d^n}/{dx^n} [(ax + b)^m] = m(m-1)(m-2)…(m-n+1)a^n (ax+b)^{m-n} $$ $$ {d^n}/{dx^n} [{1}/{(ax+b)^m}] =... View Article
Infinite Series
December 8, 2016Infinite series is denoted as: $$∑↙{n=1}↖∞ u_n$$ or $$∑u_n$$ Sum of first n terms in series is denoted by: $S_n... View Article
Hyperbolic Functions, Logarithms of Complex Numbers
December 8, 2016cosx = ${e^{ix}+e^{-ix}}/{2}$ 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... View Article
Complex Numbers
December 8, 2016z = x + iy with (x,y) as point coordinates. z = r(cosθ + isinθ), where x = r cosθ... View Article
Eigen Values, Eigen Vectors
December 8, 2016Note: $ λ_1 $ x $λ_2$ x…x $λ_n$ = |A| If $ λ_1, λ_2, …, λ_n $ are eigen values... View Article
System of Linear Algebraic Functions
December 8, 2016Condition for consistency of non-homogenous equation Consider, AX = B m = total no. of equations, n = toal no.... View Article