$\frac{d J}{d \alpha}$ or $\frac{\partial J}{\partial \alpha}$ in variational calculus

Question

We've a functional

$J(\alpha)=\int_{x_{1}}^{x_{2}} f\left\{y(\alpha, x), y^{\prime}(\alpha, x) ; x\right\} d x$

It's derivative with respect to the parameter $\alpha$ is given in textbook Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion,as $\frac{\partial J}{\partial \alpha}$

Shouldn't it have been $\frac{d J}{d \alpha}$ ? I'd read that partial notation is used when other variables are kept constant, here after the integral is performed the only variable remaining is $\alpha$ so the $\frac{d J}{d \alpha}$ notation seems appropriate. Is it true?

Answer 1

As an exercise let me deliver here these three expressions:

1. For a function $f=f(t)$ we write $$\frac{df(a)}{dt}=\lim_{t\to0}\frac{f(a+t)-f(a)}{t}$$

2. For a function on several variables $F=F(x,y)$. we have $$\frac{\partial F(a,b)}{\partial x}=\lim_{x\to0}\frac{F(a+x,b)-F(a,b)}{x}\quad {\rm and}\quad \frac{\partial F(a,b)}{\partial y}=\lim_{y\to0}\frac{F(a,b+y)-F(a,b)}{y}.$$

3. But for functional integral $J=\int_a^bL(x,y,y')dx$ we consider $$\frac{\delta J(x,y,y')}{\delta \varepsilon}= \lim_{\varepsilon\to0}\int_a^b \frac{L(x,y+\varepsilon h,y'+\varepsilon h')-L(x,y,y')}{\varepsilon}\ dx, $$ where $y=y(x)$ and $h=h(x)$ is took in such a way that satisfies $h(a)=h(b)=0$.