Total derivative

A fluid parcel is an infinitesimal, indivisible piece of fluid - effectively a very small fluid parcel of fixed mass.

The total (or material or substantive or advective or Lagrangian) derivative is the rate of change of a property of the fluid parcel.

Consider the temperature, $T$, of a parcel. The change in $T$ is given by:
$$\delta T = \frac{\partial T}{\partial t} \delta t + \frac{\partial T}{\partial x}\delta x + \frac{\partial T}{\partial y}\delta y + \frac{\partial T}{\partial z}\delta z$$

So, the change in $T$ in time $\delta t$ is:
$$\frac{\delta T}{\delta t} = \frac{\partial T}{\partial t} + \frac{\partial T}{\partial x} \frac{\delta x}{\delta t} + \frac{\partial T}{\partial y} \frac{\delta y}{\delta t} + \frac{\partial T}{\partial z} \frac{\delta z}{\delta t}$$


and so the total derivative is given by:
$$\begin{align*} \frac{DT}{dt} &= \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} \\  &= \frac{\partial T}{\partial t} + (\mathbf{v} \cdot \nabla ) T \end{align*}$$

Eulerian and Lagrangian viewpoints

In a Lagrangian point of view, we are interested in following a fluid parcel along its journey. We float along with the current.

In an Eulerian point of view, we are interested in how things change at a fixed point. We watch the flow pass by.

Laplacian

The Laplacian is the div of the grad of a function on Euclidean space:
$$\nabla^2 \psi = \nabla \cdot \nabla \psi = \Delta \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}$$

The Laplacian represents the flux density of the gradient flow of a function, and can be thought of as a measure of source or sink in the flow.

Laplace's equation is the case when the Laplacian equals zero: $\nabla^2 \rho = 0$.

Curl

The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation.
For $\mathbf{F}=(u,v,w)$: $$\nabla \times \mathbf{F} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial }{\partial y} & \frac{\partial}{\partial z} \\ u & v & w \end{array} \right| = \left( \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) \mathbf{i} + \left( \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) \mathbf{j} + \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \mathbf{k}$$
The curl gives the infinitesimal area density of the circulation of the vector field.

Divergence

The divergence of a vector field $\mathbf{F} = (u,v,w)$ is given by:
$$\nabla \cdot \mathbf{F} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}$$
Divergence measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

Gradient

The gradient of a scalar function $f(x,y,z)$ is given by:
$$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$

The gradient shows how the values of the function change. For example, if the scalar funcion is height in two dimensions, $H(x,y)$ then $\nabla  H$ is a two-dimensional vector field that points in the direction of greatest rate of increase of $H$, and whose magnitue gives the slope in that direction.

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

Del

x Del (or nabla) is an operator used in mathematics, in particular, in vector calculus, as a vector differential operator.

$$\nabla = \left( \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} \right)$$