What is the operator "capital D" and how can the chain rule be used in this way I ought to know this but I was somehow always able to avoid mixing partials and $d$s 
My book notes the following: (with some intermediate steps which are less important than the time taken to write)
Here $f$ is a function of $x(t),y(t),z(t),t$ and $u=\frac{dx}{dt}$ and "" for others.
$\frac{Df}{Dt}=\frac{\partial f}{\partial t}+(\mathbf{u}\cdot\nabla)f$
I was taught to use $D$ for total derivative but there's no malformed interpretation I could apply here. 
What is going on and if you were to look it up in an index of a book what would the book be on, and what would you look up? (E.G: Real analysis, partial differentiation)
Also I'm not sure how to use bold with LaTeX nor get a special dot. If someone could tell me in the comments I'd be very grateful. 
 A: This is just the chain rule.  It may help to think of this as follows:
We have
$f = f(x(t),y(t),z(t),w(t))$ where it so happens that $w(t) = t$.  We then have
$$
\newcommand{\pwrt}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dwrt}[2]{\frac{d #1}{d #2}}
D_tf = \pwrt fw \dwrt wt  +
\pwrt fx \dwrt xt + 
\pwrt fy  \dwrt yt + 
\pwrt fz  \dwrt zt =\\
\pwrt fw + (\mathbf u \cdot \nabla)f
$$
you may consider rewriting $\pwrt fw$ as $\pwrt ft$ a slight abuse of notation.
A: In the context of fluid/continuum mechanics, the operator $\frac{\mathrm{D}}{\mathrm{D}t}$ denotes the material derivative. This is the sense in which $\mathrm{D}$ is being used in the equation you cite in your question.
The material derivative is also often called the "total derivative", though that term is also used in a completely different sense in the context of multivariable calculus (specifically, it is the linear transformation described by the Jacobian matrix), which may be the source of your confusion.

In the study of fluid mechanics, there are two perspectives from which a physical system is commonly analysed:

*

*the Eulerian perspective, which analyses the system in a fixed (e.g. Cartesian) coordinate system $(x, y, z)$, wherein we specify properties of the system at given locations, e.g. given a velocity vector field $\mathbf{u}(x, y, z, t)$, then $\mathbf{u}(2, 4, -7, 5)$ denotes the velocity of the flow at the point $(x, y, z) = (2, 4, -7)$ at time $t=5$.


*the Lagrangian perspective, which analyses what happens to given packets/corpuscles of fluid (called fluid parcels in the literature) as the system evolves. Typically, we label each particle with its Eulerian coordinates at time $t=0$, and refer to that particle using that label at all times. The convention is to use uppercase variables $(X, Y, Z)$ to denote a Langrangian specification. For example, if a fluid parcel exists at the Eulerian position $(x, y, z) = (1, 3, 7)$ at time $t=0$, we can label that parcel $(X, Y, Z) = (1, 3, 7)$ permanently. As the fluid flow evolves over time, suppose that this parcel ends up at the Eulerian position $(x, y, z) = (2, 4, -7)$ at time $t=5$. If we denote the velocity of the parcel with Lagrangian label $(X, Y, Z)$ at time $t$ by $\mathbf{u}(X, Y, Z, t)$, then the quantity $\mathbf{u}(1, 3, 7, 5)$ (where the arguments are Lagrangian coordinates) describes precisely the same thing as the quantity $\mathbf{u}(2, 4, -7, 5)$ (where the arguments are Eulerian coordinates) mentioned in the previous bullet point. In  a given context, it should be clear whether the arguments passed to a function such as $\mathbf{u}$ are Eulerian or Lagrangian coordinates.
We can look at how any property of the system, such as the velocity $\mathbf{u}$ or the fluid density $\rho$, changes over time from either of these perspectives. With such a property denoted $\mathbf{F}$ in general, then:

*

*$\frac{\mathrm{d}\mathbf{F}}{\mathrm{d}t}$ is the regular derivative, which describes how $\mathbf{F}$ changes over time as a function of Eulerian coordinates $(x,y,z)$.


*$\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t}$ is the material derivative, which describes how $\mathbf{F}$ changes over time as a function of Lagrangian coordinates $(X,Y,Z)$. Since it is the Lagrangian analogue of the Eulerian $\frac{\mathrm{d}\mathbf{F}}{\mathrm{d}t}$, this is why the uppercase $\mathrm{D}$ is used to denote it, as per the previously mentioned convention.
If we want to express the material derivative in terms of Eulerian coordinates rather than Lagrangian coordinates, we perform a coordinate transformation from $(X,Y,Z)$ to $(x,y,z)$, which can be done using the chain rule by considering $\mathbf{F}$ for a specific fluid parcel as a function of time alone: the Lagrangian field $\mathbf{F}(t)$ for that specific fluid parcel becomes the Eurlerian field $\mathbf{F} \big( x(t), y(t), z(t), t \big)$, where $\big( x(t), y(t), z(t) \big)$ is a parameterisation of the trajectory of that fluid parcel. After applying the chain rule, everything is expressed in Eulerian coordinates, and we see that
$$\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t} \equiv \frac{\partial \mathbf{F}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{F}.$$

The above identity is often expressed as
$$\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t} \equiv \frac{\partial \mathbf{F}}{\partial t} + (\mathbf{u} \cdot \nabla) \: \mathbf{F},$$
which is the equation that you cite in your question, or as
$$\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t} \equiv \left( \frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla \right) \mathbf{F},$$
or even simply as
$$\frac{\mathrm{D}}{\mathrm{D}t} \equiv \frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla.$$
In all of these alternative ways of writing the identity, the term $\mathbf{u} \cdot \nabla$ is a specific differential operator. Some consider $\mathbf{u} \cdot \nabla$ to be abuse of notation, but regardless, it means the following: given
$$\mathbf{u}(x,y,z) \equiv \begin{pmatrix} u(x,y,z) \\ v(x,y,z) \\ w(x,y,z) \end{pmatrix},$$
then we have
$$\mathbf{u} \cdot \nabla \equiv u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z},$$
so that
$$
\begin{align}
(\mathbf{u} \cdot \nabla) \: \mathbf{F}
&\equiv \left( u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z} \right) \mathbf{F} \\
&\equiv \enspace u \frac{\partial\mathbf{F}}{\partial x} + v \frac{\partial\mathbf{F}}{\partial y} + w \frac{\partial\mathbf{F}}{\partial z} \\
&\equiv \enspace \mathbf{u} \cdot \nabla \mathbf{F}.
\end{align}
$$
Importantly, the dot operator here is not commutative, so the differential operator $\mathbf{u} \cdot \nabla$ should not be confused with the expression $\nabla \cdot \mathbf{u}$, which is the divergence of $\mathbf{u}$, and is given by
$$\nabla \cdot \mathbf{u} \equiv \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}.$$
A: It's just the chain rule.
To be more precise, write $x=X(t)$, $y=Y(t)$, $z=Z(t)$ in order to distinguish the quantities $x$, $y$, $z$ from the functions $X$, $Y$, $Z$ which say how they depend on $t$. Now take your function $f(x,y,z,t)$ and define
$$
g(t) = f(X(t),Y(t),Z(t),t)
.
$$
Then the notation $Df/Dt$ (by definition) means exactly $g'(t)$, which can be computed using the chain rule:
$$
\begin{align}
g'(t) &= f'_x(X(t),Y(t),Z(t),t) \, X'(t) \\ &+ f'_y(X(t),Y(t),Z(t),t) \, Y'(t) \\ &+ f'_z(X(t),Y(t),Z(t),t) \, Z'(t) \\ &+ f'_t(X(t),Y(t),Z(t),t),
\end{align}
$$
which equals the given expression since $X'(t)=u$, etc.
