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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 linear operators on \(E\).

ODEs

The (matrix) exponential can be used to solve certain types of first-order linear systems of ordinary differential equations with non-constant coefficients: not only can we solve \begin{align}x’(t)&=x(t)+y(t) \\ y’(t)&=x(t)+y(t),\end{align} but we can also solve \begin{align}x’(t)&=tx(t)+y(t) \\ y’(t)&=x(t)+ty(t).\end{align}

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$$
D\exp(x)u = \int_0^1 e^{sx}ue^{(1-s)x}\,ds.
$$ 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.

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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:

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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 Taylor’s theorem (covered in part 2) to prove the higher derivative test for functions on Banach spaces, and the implicit function theorem (covered in part 4) to prove a special case of the method of Lagrange multipliers.

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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.

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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 partly due to the fact that as a topological space, there is no natural injection into a product of Banach spaces.) In this post, we will look at how the existence of partial derivatives relates to differentiability, how the symmetry of higher derivatives (covered in part 2) affects mixed partial derivatives, and finally a short proof of differentiation under the integral sign.

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Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima

Last time, we covered the definition of the derivative and its basic properties, which all turn out to be quite similar to their single variable counterparts. Now we are going to explore higher derivatives. In traditional multivariable calculus, true higher derivatives do not exist (except in a specific situation which will be discussed in part 5). Of course, we have so-called “mixed/higher partial derivatives”, which are coordinate-dependent and notationally tricky to work with. As a consequence, the usual statement of Taylor’s theorem in \(\mathbb{R}^n\) ends up being ugly and hard to remember. In reality, Taylor’s theorem for Banach spaces looks almost exactly the same as the single variable Taylor’s theorem!

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Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima

In multivariable calculus, one often encounters nonsensical equations such as the following:

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Here are some series convergence problems that I gathered quite a while ago. A few of them are a bit tricky.

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