\( \newcommand{\bbc}{\mathbb{C}} \newcommand{\Int}{\operatorname{Int}} \)Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis Preliminaries Let \(F\) be a complex Banach space, let \(U\subseteq\bbc\) be an open set, and let \(f:U\to F\). Recall that the complex Fréchet derivative of \(f\) at \(z\in U\), if it […]

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis As in part 1, \(E,F\) are Banach spaces and \(U\subseteq E\) is an open set. Recall that a form \(\omega:U\to L(E,F)\) is exact if \(\omega=Df\) for some \(f:U\to F\), closed if it is differentiable […]

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis The line integral is a useful tool for working with vector fields on \(\mathbb{R}^n\), (co)vector fields on manifolds, and complex differentiable functions. However, it is often unclear how these different versions of the line […]

Last time we derived a formula for the derivative of the matrix exponential. Here we will be focusing instead on the expression $$D\exp(x)u=\exp(x)u=u\exp(x),$$ which holds whenever \(u\) commutes with \(x\). In this post, \(E\) denotes a real Banach space and \(L(E)\) denotes the space of continuous linear operators on \(E\). ODEs The (matrix) exponential can […]

$$ D\exp(x)u = \int_0^1 e^{sx}ue^{(1-s)x}\,ds. $$ This intriguing formula expresses the derivative of the exponential map on a Banach algebra as an integral. In particular, using “matrix calculus” notation we have the formula $$ d\exp(X)= \int_0^1 e^{sX}(dX)e^{(1-s)X}\,ds $$ when \(X\) is a square matrix. As we’ll see, this is not too hard to prove.

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 \(x\in(a,b)\). It is not too hard to extend this result to functions defined on more general spaces:

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In this final post, we are going to look at some applications of differentiation to locating maxima and minima of real valued functions. In order to do this, we will be using […]

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Now we’re going to prove the inverse function and implicit function theorems for Banach spaces.

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima While we saw that differentiable maps may be naturally split into component functions when the codomain is a product of Banach spaces, the situation for the domain is more complicated. (This is […]