Can you have a gradient of time?
Okay this maybe a very stupid question but in my calculus III class we introduced the gradient but I am curious why don't we also include the derivative of time in the gradient.
Thanks, math noob
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$\begingroup$This depends very heavily on the applications you are interested in. Very often the gradient is defined as
$$ \nabla \;\; =\;\; \textbf{i} \frac{\partial }{\partial x} + \textbf{j} \frac{\partial }{\partial y} + \textbf{k} \frac{\partial }{\partial z} $$
and this works fine when you're dealing with spatial distribution and 3-dimensional objects. A lot of the time in physics (and especially electrodynamics) you'll be interested in conservation laws, like the fact that current density $\textbf{J}$ and electric charge density $\rho$ are conserved spatially and temporally respectively. In other words, their relationship satisfies
$$ \nabla \cdot \textbf{J} + \frac{\partial \rho}{\partial t} \;\; =\;\; 0. $$
This gives one example of how the time derivative can play a role, but in general it's not defined explicitly with the $\nabla$ operator. Another place where the "time" derivative may take place is when you're looking at a connection (higher dimensional $\nabla$) on a manifold (generally a nonlinear space that locally looks Euclidean, on which you can do calculus). On manifolds you may easily have more than 3 variables that describe your space, and in the context of general relativity (where manifolds and time derivatives are present) your $\nabla$ operator will have a "time" derivative built into it. It's important though that when (or if) you look at higher-dimensional spaces such as manifolds you realize that the construction is made independently of the notion of time. Rather, time is just treated as another coordinate that behaves just like any other.
$\endgroup$ $\begingroup$In your Calc III class you will mostly be focusing on concepts specifically in 3 dimensions, mostly because not everybody really wants to think that hard about higher dimensional geometry at this level. Also, for many engineering applications, sticking with 3 dimensions is all you need. But it is certainly possible to do calculus, defining gradients and the like, in higher dimensions, even looking at "curved spaces", which are called manifolds. It is very common to include time as a dimension when working in relativity, and consequently the gradient will include a time derivative.
Aside from relativity, a more straightforward way a time derivative might show up is in something like the wave equation, where we are interested in, quite simply, a function of space and time, which shows the propagation of energy throughout a medium over time. So the wave function that you will be looking for is a function $f(x,y,z,t)$ which takes as its arguments the spatial position and the time and outputs, say, the energy that is propagating through a medium. In that case, you would have a straightforward gradient: $\nabla f(x,y,z,t) = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}, \frac{\partial f}{\partial t} \rangle$.
So, yes, you can have a gradient with time in it, you just probably aren't going to talk about it in your Calc III class. But you might, later on.
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