Fréchet derivative of the (matrix) exponential function
$$ D\exp(x)u = \int_0^1 e^{sx}ue^{(1-s)x}\,ds. $$ This intriguing formula expresses the derivative of the exponential map on a Banach algebra as an integral. In...
Month: February 2013
Convex functions, second derivatives and Hessian matrices
In single variable calculus, a twice differentiable function \(f:(a,b)\to\mathbb{R}\) is convex if and only if \(f^{\prime\prime}(x)\ge 0\) for all \(x\in(a,b)\...
Differentiation done correctly: 5. Maxima and minima
Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In this final post, we...
Differentiation done correctly: 4. Inverse and implicit functions
Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Now we’re going ...
Differentiation done correctly: 3. Partial derivatives
Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima While we saw that diff...
Differentiation done correctly: 2. Higher derivatives
Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Last time, we covered ...
Differentiation done correctly: 1. The derivative
Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In multivariable calcu...