# 4.1: Vector calculus operations

- Page ID
- 18060

In multidimensional calculus, the role of the derivative is taken by the vector differential operator \(\vec{\nabla}\), pronounced “del’’, or sometimes “nabla’’:

\[\nabla_{i}=\frac{\partial}{\partial x_{i}}.\]

This operator can be written in several different but equivalent ways:

\[\vec{\nabla}=\left\{\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \frac{\partial}{\partial x_{3}}\right\}=\hat{e}^{(i)} \frac{\partial}{\partial x_{i}}=\hat{e}^{(i)} \nabla_{i}.\]

## 4.1.1 Gradient of a scalar

Let \(\phi(\vec{x}\) be a scalar field and \(\vec{x}\) the position vector in a Cartesian coordinate system. Application of \(\vec{\nabla}\) yields the gradient:

\[\vec{G}=\vec{\nabla} \phi=\left\{\frac{\partial \phi}{\partial x_{1}}, \frac{\partial \phi}{\partial x_{2}}, \frac{\partial \phi}{\partial x_{3}}\right\}=\hat{e}^{(i)} \frac{\partial \phi}{\partial x_{i}}\]

The gradient has three components and appears to be a vector, but we should check. Identifying a vector is more complicated when spatial derivatives are involved.

\[\begin{equation}\begin{aligned}

&G_{i}^{\prime}=\frac{\partial \phi^{\prime}}{\partial x_{i}^{\prime}}\\

&\begin{array}{ll}

= & \frac{\partial \phi}{\partial x_{i}^{\prime}} \quad(\phi \text { is a scalar}) \\

= & \frac{\partial \phi}{\partial x_{j}} \frac{\partial x_{j}}{\partial x_{i}^{\prime}} \quad \left(\text {applying the chain rule and summing over } j\right)

\end{array}\\

&=G_{j} \frac{\partial x_{j}}{\partial x_{i}^{\prime}}.

\end{aligned} \label{eqn:1} \end{equation}\]

The partial derivative in the final term is the Jacobian matrix for the transformation. Using the reverse transformation rule Equation 3.2.7,

\[\frac{\partial x_{j}}{\partial x_{i}^{\prime}}=\frac{\partial x_{k}^{\prime} C_{j k}}{\partial x_{i}^{\prime}}=C_{j k} \frac{\partial x_{k}^{\prime}}{\partial x_{i}^{\prime}}=C_{j k} \delta_{k i}=C_{j i}.\label{eqn:2}\]

This useful result pertains to every orthogonal coordinate system, so we’ll highlight for later reference:

\[\frac{\partial x_{j}}{\partial x_{i}^{\prime}}=C_{j i}.\label{eqn:3}\]

Now, combining Equation \(\ref{eqn:1}\) and \(\ref{eqn:2}\), we have \(G^\prime_i=G_iG_{ij}\), i.e., **the gradient of a scalar transforms as a vector.**

## 4.1.2 Divergence

The divergence normally results from applying \(\vec{\nabla}\) to a vector:

\[\vec{\nabla} \cdot \vec{u}=\frac{\partial u_{i}}{\partial x_{i}}.\]

The result is a scalar. A vector whose divergence is zero is called **solenoidal**. The divergence may also be applied to either dimension of a matrix or a 2nd order tensor. For example,

\[\frac{\partial A_{i j}}{\partial x_{j}} \hat{e}^{(i)}\]

If \(\underset{\sim}{A}\) is a tensor, the result is a vector.

## 4.1.3 Curl

The curl is applied to a vector field \(\vec{u}\) by taking the cross product with \(\vec{\nabla}\):

\[\vec{\nabla} \times \vec{u}=\varepsilon_{i j k} \nabla_{i} u_{j} \hat{e}^{(k)}.\]

The result is another vector field. It can be expanded as

\[\vec{u} \times \vec{v}=\hat{e}^{(1)}\left(u_{2} v_{3}-u_{3} v_{2}\right)-\hat{e}^{(2)}\left(u_{1} v_{3}-u_{3} v_{1}\right)+\hat{e}^{(3)}\left(u_{1} v_{2}-u_{2} v_{1}\right).\label{eqn:4}\]

See Appendix D, section D.3.3 for details.

## 4.1.4 Laplacian

The Laplacian results from successive applications of \(\vec{\nabla}\):

\[\vec{\nabla} \cdot \vec{\nabla}=\nabla^{2}=\nabla_{i} \nabla_{i}=\frac{\partial^{2}}{\partial x_{i}^{2}}\]

The Laplacian may be applied to either a scalar or a vector:

\[\nabla^{2} \phi=\vec{\nabla} \cdot \vec{\nabla} \phi=\frac{\partial^{2} \phi}{\partial x_{i}^{2}},\]

or

\[\nabla^{2} \vec{u}=\left\{\nabla^{2} u_{1}, \nabla^{2} u_{2}, \nabla^{2} u_{3}\right\}=\hat{e}^{(i)} \nabla^{2} u_{i}.\]

## 4.1.5 Advective derivative

The advective derivative is an operation unique to fluid mechanics. Besides \(\vec{\nabla}\), it requires a vector field \(\vec{u}\left(\vec{x}\right)\), which is often (though not always) chosen to be the velocity of the flow. When this choice is made, we use the more common term **material derivative** (discussed in section 5.1.1). The operation is

\[[\vec{u} \cdot \vec{\nabla}]=u_{i} \frac{\partial}{\partial x_{i}}.\]

The advective derivative can be applied to a scalar, resulting in another scalar:

\[[\vec{u} \cdot \vec{\nabla}] \phi=\vec{u} \cdot \vec{\nabla} \phi=u_{i} \frac{\partial \phi}{\partial x_{i}},\]

or to a vector, resulting in another vector:

\[[\vec{u} \cdot \vec{\nabla}] \vec{v}=u_{i} \frac{\partial}{\partial x_{i}}\left(v_{j} \hat{e}^{(j)}\right)=\hat{e}^{(j)} u_{i} \frac{\partial v_{j}}{\partial x_{i}}.\]

## 4.1.6 Vector identities

Many equations hold for only certain values of a variable, and one may want to solve the equation to find those values. In contrast, *identities *are equations that hold for all values of a certain class of variables, e.g., all vectors that vary continuously in space. The identity

\[\vec{\nabla} \cdot(\vec{\nabla} \times \vec{u})=0\]

tells us that the divergence of the curl of a vector is zero, and this is true for every continuously-varying vector \(\vec{u}\). Such identities are tremendously useful in vector calculus. For example, if a term includes the divergence of the curl of a vector, you can throw it out regardless of what the vector is.

Appendix E lists 21 of the most useful vector identities. All of these can (and should) be proved using the methods we have covered so far. For example:

• Proof of identity #15: \(\vec{\nabla}\times\vec{\nabla}\phi=0\). We start with the kth component of \(\vec{\nabla}\times\vec{\nabla}\phi\):

\[\begin{aligned}

&\begin{aligned}

[\vec{\nabla} \times \vec{\nabla} \phi]_{k}=\varepsilon_{i j k} \frac{\partial}{\partial x_{i}} \frac{\partial \phi}{\partial x_{j}} &=\varepsilon_{i j k} \frac{\partial^{2} \phi}{\partial x_{i} \partial x_{j}} \\

&=\varepsilon_{i j k} \frac{\partial^{2} \phi}{\partial x_{j} \partial x_{i}}

\end{aligned} \quad \text { (reverse order of differentiation) }\\

&=\varepsilon_{j i k} \frac{\partial^{2} \phi}{\partial x_{i} \partial x_{j}} \quad \left(\text { relabel } i \text { and } j \text { as each other }\right)\\

&\begin{array}{ll}

= & -\varepsilon_{i j k} \frac{\partial^{2} \phi}{\partial x_{i} \partial x_{j}} \\

= & -[\vec{\nabla} \times \vec{\nabla} \phi]_{k}

\end{array} \quad \left( \text { use antisymmetry of } \varepsilon\right)

\end{aligned}\]

We have shown that \[[ \vec{\nabla}\times\vec{\nabla}\phi]_k\] is equal to its own additive inverse, and therefore can have no other value but zero.

• Proof of identity #21. It’s easier if we rearrange the identity like this: \( \left(\vec{\nabla}\times\vec{u}\right)\times\vec{u}\equiv \left[\vec{u}\cdot\vec{\nabla}\right]\vec{u}-\frac{1}{2}\vec{\nabla}\left(\vec{u}\cdot\vec{u}\right)\). Now define

\[\vec{\omega}=\vec{\nabla} \times \vec{u}, \quad \text { or } \quad \omega_{k}=\varepsilon_{i j k} \frac{\partial}{\partial x_{i}} u_{j}\label{eqn:5}\]

and

\[\vec{F}=(\vec{\nabla} \times \vec{u}) \times \vec{u}, \quad \text { or } \quad F_{m}=\varepsilon_{k l m} \omega_{k} u_{l}\label{eqn:6}\]

We now substitute Equation \(\ref{eqn:5}\) into Equation \(\ref{eqn:6}\) and use the \(\varepsilon - \delta\) relation:

\[\begin{aligned}

F_{m} &=\varepsilon_{k l m}\left(\varepsilon_{i j k} \frac{\partial}{\partial x_{i}} u_{j}\right) u_{l} \\

&=\varepsilon_{i j k} \varepsilon_{k l m} \frac{\partial u_{j}}{\partial x_{i}} u_{l} \\

&=\left(\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}\right) \frac{\partial u_{j}}{\partial x_{i}} u_{l} \\

&=\delta_{i l} \delta_{j m} \frac{\partial u_{j}}{\partial x_{i}} u_{l}-\delta_{i m} \delta_{j l} \frac{\partial u_{j}}{\partial x_{i}} u_{l} \\

&=\frac{\partial u_{m}}{\partial x_{l}} u_{l}-\frac{\partial u_{l}}{\partial x_{m}} u_{l}.

\end{aligned}\]

(In that last step we had two choices: we summed over the dummy indices \(i\) and \(j\), but we could just as well have chosen to sum over \(i\) and \(l\) or \(j\) and \(l\).) Now it’s just a matter of recognizing the remaining two terms on the right-hand side as the terms in the identity we want to prove.

\[\begin{aligned}

F_{m} &=u_{l} \frac{\partial}{\partial x_{l}} u_{m}-\frac{1}{2} \frac{\partial}{\partial x_{m}}\left(u_{l} u_{l}\right) \\

&=[\vec{u} \cdot \vec{\nabla}] u_{m}-\frac{1}{2} \frac{\partial}{\partial x_{m}}(\vec{u} \cdot \vec{u}),

\end{aligned}\]

or \(\vec{F}=\left[\vec{u}\cdot\vec{\nabla}\right]\vec{u}-\frac{1}{2}\vec{\nabla}\left(\vec{u}\cdot\vec{u}\right)\), and the identity is proven.

Identity-proving is not only an excellent diversion for a rainy day; it will also, like pushups for a quarterback or scales for a musician, prepare you for great things. You are therefore encouraged to see the list of vector identities in appendix E as a fine pile of puzzles awaiting your attention. **Exercise 20** lists a few that you should definitely try.