Looking at a textbook of mine, I have noticed that there is a question I've been able to answer.
Let f : $\mathbb{R}$ → $\mathbb{R}$ be bounded and continuous and define I : $\mathbb{R}$ × (0,∞) → $\mathbb{R}$ by I(x, t) := $\int_0^{x\over \sqrt4t}$ f(s) ds Calculate the Jacobian matrix of I at (x, t) ∈ $\mathbb{R}$ × (0,∞).
I'm aware that I need to find the partial derviatives of I, but I am not sure where to start considering it's an integral. Any help to get me started would be fantastic, thanks!
EDIT: I have found the partial differentials of $ {x\over\sqrt (4t)}$ to be $\partial g\over\partial x$ = $1\over 2\sqrt t$ and $\partial g\over\partial t$ = $-x\over 4t^\frac 32$. Not sure how to proceed from here.
EDIT 2: thanks for the help!
Let $F(x)=\int^{g(x)}_{h(x)} f(s)ds$ the derivative of $F$ is: \begin{align} F'(x)=g'(x) f(g(x)) - h'(x) f(h(x)) \end{align} Edit Now a function with two variables: \begin{align} F(x,t)= \int^{g(x,t)}_{h(x,t)} f(s) ds \end{align} The partial derivatives: \begin{align} \partial_x F(x,t) = \partial_x g(x,t) f(g(x,t)) - \partial_xh(x,t) f(h(x,t)) \end{align} And: \begin{align} \partial_t F(x,t) = \partial_t g(x,t) f(g(x,t)) - \partial_th(x,t) f(h(x,t)) \end{align} Now apply this to the particular case.