Finding the Jacobian matrix of an intergral

Question

How do I find the Jacobian matrix of $I : \mathbb{R} \times (0, \infty ) \rightarrow \mathbb{R}$ at $(x,t) \in \mathbb{R} \times (0, \infty) ) $ with $$I(x,t) := \int_{0}^{x(4t)^{-1/2}} f(s)\,\mathrm{d}s. $$

Answer 1

Strange, but let's try! First, we supose your $f$ is a continuous function. The Fundamental Theorem of Calculus give us:

$$\frac{\partial}{\partial t}I(x,t) = f(\sqrt(4t)) $$ The Jacobian matrix of $I(x,t)$ is

$$J[I] = [\frac{\partial I(x,t)}{\partial x} \quad \frac{\partial I(x,t)}{\partial t}] $$

So $$J[I] = [\frac{\partial I(x,t)}{\partial x} \quad f(\sqrt(4t)) ] $$

But $I$ doens`t depends of $x$, so $\frac{\partial I(x,t)}{\partial x} = 0$

So we have $$J[I] = [0 \quad f(\sqrt(4t)) ] .$$