differentiation integral symbol I hopefully have a question which should be obvious to all but has been bothering me for sometime.

I have used the symbol $d$ for the differential entity of integration, for example, $$\int{x}{dx}=\frac{1}{2}x^2+c$$

Common to partial differential equations, I have typically used $d$ or $D$ to represent the total (or material or convective...) derivative as opposed to using $\partial$ to represent the partial derivative of a value.

My question is: why is $d$ still typically used when integrating partial differential equations, when in fact, I would argue that I am integrating with respect to a partial differential?

As an example, shouldn't $$\int_{x_{0}}^{x_{1}} \frac{\partial f(x,y)}{\partial x} dx$$
be written as $$\int_{x_{0}}^{x_{1}} \frac{\partial f(x,y)}{\partial x} \partial x$$
especially since I am only integrating over $x$?

Am I missing something, or is this just overlooked?

Thanks,
Jon
First, the derivative should always be written using an upright 'd' as in

$$ \int f(x) \mathrm{d}x $$

To address your question regarding the integration of a multivariable function, the integration can be seen as the integral with respect to each variable, just over a singleton bound.

$$ \int f(x,y)\mathrm{d}x = \int_{x_0}^{x_1}\int_{y_0}^{y_0}f(x,y) \mathrm{d}y \mathrm{d}x $$

So, are you doing a 'partial' integration? Not really. When you integrate only with respect to one variable, you're just integrating with respect to the second variable as well, but over an infinitesimal bound.
First, the derivative should always be written using an upright 'd' as in

$$ \int f(x) \mathrm{d}x $$

To address your question regarding the integration of a multivariable function, the integration can be seen as the integral with respect to each variable, just over a singleton bound.

$$ \int f(x,y)\mathrm{d}x = \int_{x_0}^{x_1}\int_{y_0}^{y_0}f(x,y) \mathrm{d}y \mathrm{d}x $$

So, are you doing a 'partial' integration? Not really. When you integrate only with respect to one variable, you're just integrating with respect to the second variable as well, but over an infinitesimal bound.