Vector Calculus Notes

Last reviewed 16 Sept 2026 · 7 min read

Scalar and vector fields

Scalar field — a scalar at each point, e.g. temperature $T (x, y, z)$ , pressure, potential.
Vector field — a vector at each point, e.g. velocity of fluid $V (x, y, z)$ , force field, electric field.

Del operator:

\nabla = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z}

For a scalar field $ϕ$ :

\nabla ϕ = \frac{\partial ϕ}{\partial x} \hat{i} + \frac{\partial ϕ}{\partial y} \hat{j} + \frac{\partial ϕ}{\partial z} \hat{k}

$\nabla ϕ$ is a vector normal (perpendicular) to the level surface $ϕ = c$ .
It points in the direction of the maximum rate of increase of $ϕ$ ; the maximum rate is $∣\nabla ϕ ∣$ .
Unit normal to surface $ϕ = c$ : $\overset{n}{^} = \frac{\nabla ϕ}{∣\nabla ϕ ∣}$ .

FormulaDirectional derivative

Rate of change of $ϕ$ in the direction of vector $a$ :

D_{\overset{a}{^}} ϕ = \nabla ϕ \cdot \overset{a}{^} = \frac{\nabla ϕ \cdot a}{∣ a ∣}

Maximum value $= ∣\nabla ϕ ∣$ (along $\nabla ϕ$ ); zero along directions tangent to the level surface.

Angle between surfaces at a point = angle between their normals (gradients).

For $F = F_{1} \hat{i} + F_{2} \hat{j} + F_{3} \hat{k}$ :

\nabla \cdot F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}

A scalar — the net outflow (flux) per unit volume at a point (source if positive, sink if negative).
$\nabla \cdot F = 0$ → solenoidal field (incompressible fluid flow — continuity equation $\nabla \cdot V = 0$ ).

\nabla \times F = \hat{i} \frac{\partial}{\partial x} F_{1} \hat{j} \frac{\partial}{\partial y} F_{2} \hat{k} \frac{\partial}{\partial z} F_{3}

A vector measuring rotation (circulation per unit area); for fluid flow, curl $V = 2 ω$ (twice the angular velocity) = vorticity.
$\nabla \times F = 0$ → irrotational field → $F = \nabla ϕ$ for some scalar potential $ϕ$ → conservative field.

\nabla^{2} ϕ = \nabla \cdot \nabla ϕ = \frac{\partial ^{2} ϕ}{\partial x ^{2}} + \frac{\partial ^{2} ϕ}{\partial y ^{2}} + \frac{\partial ^{2} ϕ}{\partial z ^{2}}

Laplace's equation $\nabla^{2} ϕ = 0$ — harmonic functions (potential flow, steady heat conduction, seepage flow nets).

Identity	Meaning
$\nabla \times (\nabla ϕ) = 0$	Curl of gradient is zero — gradient fields are irrotational
$\nabla \cdot (\nabla \times F) = 0$	Divergence of curl is zero — curl fields are solenoidal
$\nabla \cdot (ϕ F) = ϕ \nabla \cdot F + F \cdot \nabla ϕ$	Product rule for divergence
$\nabla \times (ϕ F) = ϕ \nabla \times F + \nabla ϕ \times F$	Product rule for curl
$\nabla \times (\nabla \times F) = \nabla (\nabla \cdot F) - \nabla^{2} F$	Curl of curl
$\nabla r^{n} = n r^{n - 2} r$ ; $\nabla \cdot r = 3$ ; $\nabla \times r = 0$	For position vector $r$ , $r = ∣ r ∣$

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