multivar-calc.md (3452B)
1 --- 2 title: "Multivariable Calculus Cheatsheet" 3 date: 2017-11-14 4 tags: math calculus cheatsheet 5 categories: math 6 math: true 7 --- 8 9 # trig identities 10 11 $$\sin^2 \theta + \cos^2 \theta = 1$$ 12 13 $$1+ \tan^2 \theta = \sec^2 \theta$$ 14 15 $$1+ \cot^2 \theta = \csc^2 \theta$$ 16 17 $$\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)$$ 18 19 $$\cos(a \pm b) = \cos(a)\cos(b) \pm \sin(a)\sin(b)$$ 20 21 $$\sin(-\theta) = -\sin \theta$$ 22 23 $$\cos(-\theta) = \cos \theta$$ 24 25 $$\tan(-\theta) = -\tan \theta$$ 26 27 # derivatives 28 29 $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ 30 31 $$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\frac{dy}{dx}}{\frac{dx}{dt}}$$ 32 33 $$\frac{d}{d\theta}\sin \theta = \cos \theta$$ 34 35 $$\frac{d}{d\theta}\cos \theta = -\sin \theta$$ 36 37 $$\frac{d}{d\theta}\tan \theta = \sec^2 \theta$$ 38 39 $$\frac{d}{d\theta}\cot \theta = -\csc^2 \theta$$ 40 41 $$\frac{d}{d\theta}\sec \theta = \sec \theta \tan \theta$$ 42 43 $$\frac{d}{d\theta}\csc \theta = -\csc \theta \cot \theta$$ 44 45 $$\frac{d}{dx}\sin^{-1} x = \frac1{\sqrt{1-x^2}}, x \in [-1,1]$$ 46 47 $$\frac{d}{dx}\cos^{-1} x = \frac{-1}{\sqrt{1-x^2}}, x \in [-1,1]$$ 48 49 $$\frac{d}{dx}\tan^{-1} x = \frac1{1+x^2}, x \in [\frac{-\pi}2, \frac{\pi}2]$$ 50 51 $$\frac{d}{dx}\sec^{-1} x = \frac1{|x|\sqrt{x^2-1}}, |x| > 1$$ 52 53 # integrals 54 55 $$\int{udv} = uv - \int{vdu}$$ 56 57 $$\int_a^b{f(x)dx} = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\Delta x$$ 58 59 $$\iint_R{f(x,y)dA} = \lim_{\substack{n \to \infty \\ m \to \infty}} \sum_{i=1}^n \sum_{j=1}^m f(x_i,y_i)\Delta x\Delta y$$ 60 61 # vectors 62 63 $$\vec a \cdot \vec b = |\vec a||\vec b| \cos \theta$$ 64 65 $$|\vec a \times \vec b| = |\vec a||\vec b| \sin \theta$$ 66 67 $$\mathrm{proj}_{\vec b} \vec a = \frac{\vec a \cdot \vec b}{|\vec b|}\frac{\vec b}{|\vec b|}$$ 68 69 $$\cos \theta = \frac{n_1n_2}{|n_1||n_2|}$$ 70 71 $$D = \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}} = \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$ 72 73 $$A = \int_\alpha^\beta g(t)f'(t)dt$$ 74 75 $$L = \int_\alpha^\beta\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$ 76 77 $$a_T(t) = \frac{r'(t) \cdot r'(t)'}{|r'(t)|}$$ 78 79 $$a_N(t) = \frac{|r'(t) \times r'(t)'|}{|r'(t)|}$$ 80 81 # surfaces 82 83 | name | equation | 84 | ----------------------- | ---------------------------------------------------------- | 85 | ellipsoid | $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ | 86 | elliptical paraboloid | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$ | 87 | hyperbolic paraboloid | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$ | 88 | cone | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$ | 89 | hyperboloid of 1 sheet | $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ | 90 | hyperboloid of 2 sheets | $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ | 91 92 # polar coordinates 93 94 $$x = r \cos \theta$$ 95 96 $$y = r \sin \theta$$ 97 98 $$r = \sqrt{x^2+y^2}$$ 99 100 $$\tan \theta = \frac{y}{x}$$ 101 102 # curvature 103 104 $$\vec T(t) = \frac{\vec r'{t}}{|r'(t)|}$$ 105 106 $$\vec N(t) = \frac{\vec T'{t}}{|T'(t)|}$$ 107 108 $$\vec B(t) = \vec T(t) \times \vec N(t)\$$ 109 110 $$\kappa(t) = \left| \frac{d\vec T}{ds} \right| = \frac{|\vec T(t)|}{|\vec r(t)|} = \frac{|r'(t) \times r''(t)|}{|r'(t)|^3} = \frac{|f''(x)|}{(1+f'(x)^2)^{\frac{3}{2}}}$$ 111 112 # partial derivatives 113 114 $$z - z_0 = \frac{\partial z}{\partial x}(x-a) + \frac{\partial z}{\partial y}(y-b)$$ 115 116 $$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy$$ 117 118 $$D(a,b) = f_{xx}(a,b)f_{yy}(a,b)-f_{xy}(a,b)^2$$