Vector Calculus Preliminaries

Chapter 5: Integral Theorems of Vector Calculus

Learning objectives

Compute gradient, divergence, and curl of vector fields in $\mathbb{R}^3$
Read the geometric meaning of $\nabla$ , $\nabla\cdot$ , and $\nabla\times$ off the symbol
Use the second-derivative identities $\nabla\times(\nabla f)=\mathbf{0}$ and $\nabla\cdot(\nabla\times\mathbf{F})=0$
Recognise where these operators appear in fluid mechanics, electromagnetism, and optimisation

Vector calculus is the bridge between point-wise differential reasoning and global integral statements about flow, flux, and circulation. Before we can state the Divergence Theorem or Stokes' Theorem, we need three differential operators that act on functions and vector fields in $\mathbb{R}^3$ . Each one has a precise algebraic definition, but more importantly each one tells a geometric story: the gradient points uphill, the divergence measures expansion, and the curl measures rotation. Once you can read these meanings off the symbols, the integral theorems stop looking like formulas to memorise and start looking like conservation laws to apply.

The gradient

For a scalar function $f(x,y,z)$ , the gradient is the vector field

\nabla f = \left(\dfrac{\partial f}{\partial x},\; \dfrac{\partial f}{\partial y},\; \dfrac{\partial f}{\partial z}\right).

$\nabla f$ points in the direction of steepest increase of $f$ , and its magnitude is the rate of increase in that direction. Perpendicular to $\nabla f$ , the function is locally constant — this is why level surfaces of $f$ are everywhere perpendicular to $\nabla f$ .

Divergence

For a vector field $\mathbf{F}=(F_1, F_2, F_3)$ , the divergence is the scalar

\nabla \cdot \mathbf{F} = \dfrac{\partial F_1}{\partial x} + \dfrac{\partial F_2}{\partial y} + \dfrac{\partial F_3}{\partial z}.

The divergence at a point measures the net outward flux per unit volume of an infinitesimal box centred there. Positive divergence: a source. Negative: a sink. Zero everywhere: incompressible.

Curl

The curl of $\mathbf{F}$ is the vector field

\nabla \times \mathbf{F} = \left(\dfrac{\partial F_3}{\partial y} - \dfrac{\partial F_2}{\partial z},\; \dfrac{\partial F_1}{\partial z} - \dfrac{\partial F_3}{\partial x},\; \dfrac{\partial F_2}{\partial x} - \dfrac{\partial F_1}{\partial y}\right).

The curl at a point is twice the angular velocity of an infinitesimal paddle wheel placed in the flow. Its direction is the rotation axis (by the right-hand rule). Zero curl everywhere: irrotational.

Two second-derivative identities you will use everywhere

For any $C^2$ function $f$ : $\nabla\times(\nabla f) = \mathbf{0}$ — the curl of a gradient is zero. For any $C^2$ field $\mathbf{F}$ : $\nabla\cdot(\nabla\times\mathbf{F}) = 0$ — the divergence of a curl is zero. Both follow from equality of mixed partial derivatives. In chapter 6 you will see these are both manifestations of one structural fact: $d^2 = 0$ for differential forms.

Where this shows up

- **Fluid mechanics:** the continuity equation

\partial\rho/\partial t + \nabla\cdot(\rho\mathbf{v}) = 0

is conservation of mass written using divergence. An incompressible flow is exactly one with

\nabla\cdot\mathbf{v}=0

. - **Electromagnetism:** Maxwell's equations in differential form are four statements about

\nabla\cdot

and

\nabla\times

applied to

\mathbf{E}

and

\mathbf{B}

. Knowing these operators IS knowing electromagnetism's local structure. - **Atmospheric science:** the curl of the wind field gives *vorticity*, which controls hurricane formation, jet-stream stability, and the rotation of weather systems on rotating planets. - **Optimisation and ML:** gradient descent literally moves along

-\nabla L

for a loss

L

. Every modern neural network is trained by computing gradients of a scalar loss with respect to millions of parameters.

Pause and think: If $\mathbf{F}=\nabla f$ for some potential $f$ , what does that force the curl of $\mathbf{F}$ to be? Conversely, if you encounter a field whose curl is nonzero, can it possibly be a gradient?

Try it

Predict first, then compute: for $\mathbf{F}(x,y,z) = (x^2, y^2, z^2)$ , is the divergence at $(1,1,1)$ positive, negative, or zero? Then compute it.
For $f(x,y,z) = x^2 + y^2 + z^2$ , compute $\nabla f$ at $(1,2,3)$ . Where does it point? (Answer in words.)
For $\mathbf{F} = (-y, x, 0)$ (a rotational flow about the $z$ -axis), predict the direction of $\nabla\times\mathbf{F}$ before computing. Then compute and check.
True or false: every vector field $\mathbf{F}$ with $\nabla\times\mathbf{F}=\mathbf{0}$ can be written $\mathbf{F}=\nabla f$ for some $f$ . (Hint: the answer depends on the topology of the domain. We will revisit this in §6.2.)

A trap to watch for

Beginners often write $\nabla\cdot\mathbf{F}$ when they mean $\nabla f$ , or use a dot where they should use a cross. Anchor on the shapes: $\nabla f$ takes a scalar to a vector, $\nabla\cdot\mathbf{F}$ takes a vector to a scalar, $\nabla\times\mathbf{F}$ takes a vector to a vector. If the output type does not match what you wrote, you have used the wrong operator.

What you now know

You can compute $\nabla f$ , $\nabla\cdot\mathbf{F}$ , and $\nabla\times\mathbf{F}$ symbolically; you can read each one as a geometric statement; and you know the two identities that drop out whenever an operator is iterated. Next we use these to state the Divergence Theorem (§5.2) and Stokes' Theorem (§5.3), then unpack their physical meaning (§5.4) and proof strategies (§5.5).

References

Garrity, T. (2002). All the Mathematics You Missed. Cambridge University Press, ch. 5.
Marsden, J. E., Tromba, A. J. (2011). Vector Calculus (6th ed.). W. H. Freeman.
Schey, H. M. (2004). Div, Grad, Curl, and All That (4th ed.). W. W. Norton.
Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Cambridge UP, ch. 1.
Spivak, M. (1965). Calculus on Manifolds. W. A. Benjamin, ch. 4.