Calculus
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Jacobian Matrix
Suppose \(f: \R^n \to \R^m\), with input \(x \in \R^n\) and output \(y \in \R^m\): \[ f = \begin{cases} y_1 = f_1(x_1, x_2, ..., x_n) \\ y_2 = f_2(x_1, x_2, .
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Spherical Coordinates
The conversion between the 2-d Cartesian coordinate system and the 2-d polar coordinate system can be extended to a higher dimension, say \(k\)-d. In \(k\)-d case, their conversion can be
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Lipschitz Continuity
For a continuous mapping \(f\), it is \(K\)-Lipschitz continuous if there exists a number \(K\) such that \(\forall x,y \in \dom(f)\) \[ ||f(x) - f(y)|| \le K||x - y|| \] If the gradient of \(f\) is \(K\)-Lipschitz continuous, we further say \(f\) is \(K\)-smooth.