Function not an explicit function of time notation So far, in multivariable calculus, I know that given a function $\mathcal{H}\left(\mathbf{x},t\right)$, where $\mathbf{x} = \begin{bmatrix}x_1 & x_2 & \dots & x_n\end{bmatrix}^\textrm{T}$ is also a function of time, the total derivative is
$$\frac{d\mathcal{H}}{dt} = \frac{\partial\mathcal{H}}{\partial t} + \sum_{i=1}^n\frac{\partial\mathcal{H}}{\partial x_i}\frac{d x_i}{d t}$$
If $\mathcal{H}$ doesn't depend explicitly on time (that is, no direct relationship), then $\partial\mathcal{H}/\partial t = 0$. Can one say that
$$\mathcal{H} \neq \mathcal{H}\left(t\right)$$ or 
$$\mathcal{H} \neq \mathcal{f}\left(t\right)?$$
In mathematical notation, which one is more appropriate?
 A: No, writing $\mathcal{H}\neq\mathcal{H}(t)$ makes no sense. Just write $\partial\mathcal{H}/\partial t=0$, which conveys exactly what it says: $\mathcal{H}$ does not change when $t$ changes.
A: A note on notation
In general, when using "$f(x)$" function notation, $f$ is used to denote the function itself (e.g. $\lambda x.fx$), whereas $f(x)$ denotes the value of $f$ at $x$. 
It is technically true that $\mathcal{H}\ne\mathcal{H}(t)$, as "the function $\mathcal{H}$" is not equal to "the value of $\mathcal{H}$ at $t$". This is not especially informative, though, since $t$ can just as easily be replaced with a picture of a fish and the statement would remain true.
As stated in obscuran's answer, in order to say that $\mathcal{H}$ is independent of $t$ you would write $\partial\mathcal{H}/\partial t=0$.
Now, if you want to use $\mathcal{H}\ne\mathcal{H}(t)$ as shorthand in your own personal notes, that's fine. You don't have to stick to any particular format if you are only writing a note to yourself - just be sure to use proper notation on assignments / in publications.
