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29 Cards in this Set
- Front
- Back
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Cauchy-Shwartz Inequality |
|x dot y| <=||x||* ||y|| Are equal when x = cy |
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Dot product |
Scalar = a1b1 + a2b2 .... + anbn ||a||^2 = a dot a Take dot product of two perpendicular vectors = 0 commutative ||a||*||b||cos(theta) |
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Triangle Inequality |
||x + y|| <= ||x|| + ||y|| |
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Angle between vectors |
(a dot b) = ||a||* ||b||* cos(theta) |
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Determinant [ a b c d] |
det = ad - bc |
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Jacobian Matrix |
Df(x,y) = [df1/dx df1/dy df2/dx df2/y] |
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Chain Rule |
D(f o g)(xo, yo) = Dfg(xo,yo) * Dg(xo,yo) Rn to Rm to Rk nxk matrix = kxm matrix times mxn matrix |
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Distance point to plane |
|Ax + By + Cz +D| / (sqrt(A^2 + B^2 +C^2)) (p - b) dot n / ||n|| b is found with equation of the plane |
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Equation of a plane |
Ax + By + Cz + D = 0 Normal vector (A,B,C) A is slope of tangent vector in x-z plane B is slope of tangent vector in y-z plane |
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Cross Product |
cover up the one that you are trying to find, take det of remaining det1 - det 2 + det3 a x b = 0 if they are colinear |
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j X k k X i i X j crossed with itself |
j X k = i k X i = j i X j = k (flipped negative) crossed with itself = 0 |
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Partial derivatives |
df/dx(x,y) = lim h to 0: (f(x+h, y) - f(x,y)) / h df/dy(x,y) = lim h to 0: (f(x, y+h) - f(x,y)) / h |
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equation of tangent plane |
gradient dot eqn of a plane |
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gradient |
(df/dx, df/dy, df/dz) normal to the tangent plane points along the direction where f is increasing the fastes perpendicular to level set |
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iterated partials: d^2f/dxdy d^2f/dydx fxy fyx |
d^2f/dxdy = take x differential of y derivative d^2f/dydx = take y differential of x derivative fxy = take y differential of x derivative fyx = take x differential of y derivative mixed partials equal each other |
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Hessian matrix |
matrix that contains all second partial derivatives of function top row with x first, bottom row with y first |
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Equation of the tangent line |
l(t) = p + v(t) p = point v = perpendicular vector |
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How to find a perpendicular vector |
Ex: (2,4) Dot product must = 0, so 2x + 4y = 0 Arbitrarily find an x and y Make it so that when you take the dot product both are 0 |
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C1 and C2 |
Continuous and its derivatives are continuous |
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Directional derivative |
gradient of f(a) dotted with v (unit vector) = a real number lim h to 0: (f(x+hei) - f(x)) / h ordinary derivative of a composite function |
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Good linear approximation |
f(a) + f'(a) (t-a) |
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Multiply matrices |
sum of first row and second column; each element multiplied by the corresponding element in the other matrix # row A = # columns B |
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Boundary |
Small ball epsilon around point where some lie in A and some do not |
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Closed/Open sets |
Closed: contains all its boundary points Open: contains none of its boundary points (some epsilon neighborhood entirely contained in a) |
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Epsilon/Delta definition |
limit as x goes to a of f(x) = b, for any epsilon >0, there exists a delta >0, so that |x-a| < d and |f(x)-b| < E |
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Component functions |
If components are continuous, so is function |
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Curves/Paths |
Curve is one dimensional path parametrizes the curve |
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Differentiable if |
Df(xo) exists and lim x to xo ||f(x)- Df(xo)(x-xo) - f(xo)|| / ||x-xo|| = 0 |
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Local min/max |
a is a critical point if gradient of f(a) = 0 |
