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commit e9a758dbf6b823889d0ddd1bc46ea17785af378a
parent 7a1747568e321dfe10062c2203012e05a5558fdd
Author: Ed van Bruggen <ed@edryd.org>
Date:   Tue, 14 Nov 2017 21:18:24 -0800

Publish multivariable calc cheatsheet

Diffstat:
_posts/2017-11-14-multivar-calc.md | 117+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 117 insertions(+), 0 deletions(-)

diff --git a/_posts/2017-11-14-multivar-calc.md b/_posts/2017-11-14-multivar-calc.md @@ -0,0 +1,117 @@ +--- +title: "multivariable calculus cheatsheet" +tags: math calculus cheatsheet +categories: math +math: true +--- + +# trig identities + +$$\sin^2 \theta + \cos^2 \theta = 1$$ + +$$1+ \tan^2 \theta = \sec^2 \theta$$ + +$$1+ \cot^2 \theta = \csc^2 \theta$$ + +$$\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)$$ + +$$\cos(a \pm b) = \cos(a)\cos(b) \pm \sin(a)\sin(b)$$ + +$$\sin(-\theta) = -\sin \theta$$ + +$$\cos(-\theta) = \cos \theta$$ + +$$\tan(-\theta) = -\tan \theta$$ + +# derivatives + +$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ + +$$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\frac{dy}{dx}}{\frac{dx}{dt}}$$ + +$$\frac{d}{d\theta}\sin \theta = \cos \theta$$ + +$$\frac{d}{d\theta}\cos \theta = -\sin \theta$$ + +$$\frac{d}{d\theta}\tan \theta = \sec^2 \theta$$ + +$$\frac{d}{d\theta}\cot \theta = -\csc^2 \theta$$ + +$$\frac{d}{d\theta}\sec \theta = \sec \theta \tan \theta$$ + +$$\frac{d}{d\theta}\csc \theta = -\csc \theta \cot \theta$$ + +$$\frac{d}{dx}\sin^{-1} x = \frac1{\sqrt{1-x^2}}, x \in [-1,1]$$ + +$$\frac{d}{dx}\cos^{-1} x = \frac{-1}{\sqrt{1-x^2}}, x \in [-1,1]$$ + +$$\frac{d}{dx}\tan^{-1} x = \frac1{1+x^2}, x \in [\frac{-\pi}2, \frac{\pi}2]$$ + +$$\frac{d}{dx}\sec^{-1} x = \frac1{|x|\sqrt{x^2-1}}, |x| > 1$$ + +# integrals + +$$\int{udv} = uv - \int{vdu}$$ + +$$\int_a^b{f(x)dx} = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\Delta x$$ + +$$\iint_R{f(x,y)dA} = \lim_{\substack{n \to \infty \\ m \to \infty}} \sum_{i=1}^n \sum_{j=1}^m f(x_i,y_i)\Delta x\Delta y$$ + +# vectors + +$$\vec a \cdot \vec b = |\vec a||\vec b| \cos \theta$$ + +$$|\vec a \times \vec b| = |\vec a||\vec b| \sin \theta$$ + +$$\mathrm{proj}_{\vec b} \vec a = \frac{\vec a \cdot \vec b}{|\vec b|}\frac{\vec b}{|\vec b|}$$ + +$$\cos \theta = \frac{n_1n_2}{|n_1||n_2|}$$ + +$$D = \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}} = \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$ + +$$A = \int_\alpha^\beta g(t)f'(t)dt$$ + +$$L = \int_\alpha^\beta\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$ + +$$a_T(t) = \frac{r'(t) \cdot r'(t)'}{|r'(t)|}$$ + +$$a_N(t) = \frac{|r'(t) \times r'(t)'|}{|r'(t)|}$$ + +# surfaces + +| name | equation | +| ----------------------- | ---------------------------------------------------------- | +| ellipsoid | $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ | +| elliptical paraboloid | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$ | +| hyperbolic paraboloid | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$ | +| cone | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$ | +| hyperboloid of 1 sheet | $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ | +| hyperboloid of 2 sheets | $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ | + +# polar coordinates + +$$x = r \cos \theta$$ + +$$y = r \sin \theta$$ + +$$r = \sqrt{x^2+y^2}$$ + +$$\tan \theta = \frac{y}{x}$$ + +# curvature + +$$\vec T(t) = \frac{\vec r'{t}}{|r'(t)|}$$ + +$$\vec N(t) = \frac{\vec T'{t}}{|T'(t)|}$$ + +$$\vec B(t) = \vec T(t) \times \vec N(t)\$$ + +$$\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}}}$$ + +# partial derivatives + +$$z - z_0 = \frac{\partial z}{\partial x}(x-a) + \frac{\partial z}{\partial y}(y-b)$$ + +$$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy$$ + +$$D(a,b) = f_{xx}(a,b)f_{yy}(a,b)-f_{xy}(a,b)^2$$