Vector Analysis

UPSC Mathematics Optional · Paper I · Grad, Div, Curl & Integral Theorems

1. Vector Algebra 2. Gradient, Divergence, Curl 3. Vector Identities 4. Line Integrals 5. Surface Integrals 6. Green's Theorem 7. Stokes' Theorem 8. Gauss's Divergence Theorem 9. Conservative Fields & Potentials 10. Curvilinear Coordinates

§1 · Vector Algebra

Products PYQ★★

Dot: \(\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta\). Cross: \(|\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta\), direction by RHR.

Scalar triple product: \([\mathbf{a},\mathbf{b},\mathbf{c}]=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}\)

= volume of parallelepiped. Cyclic: \([\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=[\mathbf{b}\,\mathbf{c}\,\mathbf{a}]=[\mathbf{c}\,\mathbf{a}\,\mathbf{b}]\).

Vector triple product: \(\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=(\mathbf{a}\cdot\mathbf{c})\mathbf{b}-(\mathbf{a}\cdot\mathbf{b})\mathbf{c}\) ("BAC-CAB rule").

§2 · Gradient, Divergence, Curl

Del Operator \(\nabla\)

\(\nabla = \mathbf{i}\dfrac{\partial}{\partial x}+\mathbf{j}\dfrac{\partial}{\partial y}+\mathbf{k}\dfrac{\partial}{\partial z}\)

Gradient PYQ★★★

\(\text{grad}\,\phi = \nabla\phi = \left(\dfrac{\partial\phi}{\partial x},\dfrac{\partial\phi}{\partial y},\dfrac{\partial\phi}{\partial z}\right)\)

Points in direction of max rate of increase of \(\phi\).
Magnitude \(|\nabla\phi|\) = max rate of change.
Directional derivative: \(D_{\hat u}\phi = \nabla\phi\cdot\hat u\).
\(\nabla\phi\) is normal to the level surface \(\phi=\text{const}\).
Unit normal to surface \(F=0\): \(\hat n=\dfrac{\nabla F}{|\nabla F|}\).

Divergence PYQ★★★

\(\text{div}\,\mathbf{F}=\nabla\cdot\mathbf{F}=\dfrac{\partial F_1}{\partial x}+\dfrac{\partial F_2}{\partial y}+\dfrac{\partial F_3}{\partial z}\)

Physical meaning: net outward flux per unit volume. Source if \(>0\), sink if \(<0\).

\(\mathbf{F}\) is solenoidal iff \(\nabla\cdot\mathbf{F}=0\).

Curl PYQ★★★

\(\text{curl}\,\mathbf{F}=\nabla\times\mathbf{F}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\partial_x&\partial_y&\partial_z\\F_1&F_2&F_3\end{vmatrix}\)

Physical meaning: rotation/circulation per unit area.

\(\mathbf{F}\) is irrotational iff \(\nabla\times\mathbf{F}=\mathbf{0}\).

Laplacian

\(\nabla^2\phi = \nabla\cdot(\nabla\phi) = \dfrac{\partial^2\phi}{\partial x^2}+\dfrac{\partial^2\phi}{\partial y^2}+\dfrac{\partial^2\phi}{\partial z^2}\)

\(\phi\) is harmonic iff \(\nabla^2\phi=0\).

§3 · Vector Identities

Key Identities PYQ★★★

\(\nabla(\phi\psi)=\phi\nabla\psi+\psi\nabla\phi\)
\(\nabla\cdot(\phi\mathbf{F})=\phi\nabla\cdot\mathbf{F}+\mathbf{F}\cdot\nabla\phi\)
\(\nabla\times(\phi\mathbf{F})=\phi(\nabla\times\mathbf{F})+(\nabla\phi)\times\mathbf{F}\)
\(\nabla\cdot(\mathbf{A}\times\mathbf{B})=\mathbf{B}\cdot(\nabla\times\mathbf{A})-\mathbf{A}\cdot(\nabla\times\mathbf{B})\)
\(\nabla\times(\mathbf{A}\times\mathbf{B})=\mathbf{A}(\nabla\cdot\mathbf{B})-\mathbf{B}(\nabla\cdot\mathbf{A})+(\mathbf{B}\cdot\nabla)\mathbf{A}-(\mathbf{A}\cdot\nabla)\mathbf{B}\)
\(\nabla\times(\nabla\phi)=\mathbf{0}\) (curl of gradient is zero)
\(\nabla\cdot(\nabla\times\mathbf{F})=0\) (div of curl is zero)
\(\nabla\times(\nabla\times\mathbf{F})=\nabla(\nabla\cdot\mathbf{F})-\nabla^2\mathbf{F}\)
\(\nabla(r^n)=nr^{n-2}\mathbf{r}\). \(\nabla^2(r^n)=n(n+1)r^{n-2}\). \(\nabla^2(1/r)=0\) for \(r\ne 0\).
\(\nabla\cdot\mathbf{r}=3\). \(\nabla\times\mathbf{r}=\mathbf{0}\). \(\nabla r=\hat r=\mathbf{r}/r\).

§4 · Line Integrals

Line Integral PYQ★★★

\(\displaystyle\int_C\mathbf{F}\cdot d\mathbf{r}=\int_C(F_1\,dx+F_2\,dy+F_3\,dz)\)

Parametrize \(C\): \(\mathbf{r}=\mathbf{r}(t)\), then \(d\mathbf{r}=\mathbf{r}'(t)\,dt\).

Work done by force \(\mathbf{F}\) along \(C\) = \(\int_C\mathbf{F}\cdot d\mathbf{r}\).

§5 · Surface Integrals

Flux Integral PYQ★★★

\(\displaystyle\iint_S\mathbf{F}\cdot d\mathbf{S}=\iint_S\mathbf{F}\cdot\hat n\,dS\)

For surface \(z=g(x,y)\): \(d\mathbf{S}=(-g_x,-g_y,1)\,dx\,dy\) (upward normal).

For surface \(F(x,y,z)=0\): \(\hat n=\dfrac{\nabla F}{|\nabla F|}\), \(dS=\dfrac{dA}{|\hat n\cdot\hat k|}\) where \(dA\) is projected area element.

§6 · Green's Theorem

Green's Theorem in the Plane PYQ★★★

\[\oint_C(M\,dx+N\,dy)=\iint_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dA\]

\(C\) = positively oriented (counterclockwise) boundary of \(D\).

Area formula: \(A=\dfrac{1}{2}\oint_C(x\,dy-y\,dx)\).

Green's first identity: \(\iint_D\phi\nabla^2\psi\,dA=\oint_C\phi\dfrac{\partial\psi}{\partial n}\,ds-\iint_D\nabla\phi\cdot\nabla\psi\,dA\)

§7 · Stokes' Theorem

Stokes' Theorem PYQ★★★

\[\oint_C\mathbf{F}\cdot d\mathbf{r}=\iint_S(\nabla\times\mathbf{F})\cdot d\mathbf{S}\]

\(C\) = positively oriented boundary of surface \(S\) (right-hand rule: curl fingers along \(C\), thumb points in \(\hat n\) direction).

Consequence: For irrotational \(\mathbf{F}\) (\(\nabla\times\mathbf{F}=0\)): line integral around any closed curve = 0, integral is path-independent.

§8 · Gauss's Divergence Theorem

Gauss Divergence Theorem PYQ★★★

\[\oiint_S\mathbf{F}\cdot d\mathbf{S}=\iiint_V\nabla\cdot\mathbf{F}\,dV\]

\(S\) = closed surface bounding volume \(V\), outward normal.

Applications: \(\oiint_S\hat n\,dS=\mathbf{0}\). \(\oiint_S\mathbf{r}\cdot d\mathbf{S}=3V\). \(\oiint_S\mathbf{F}\times d\mathbf{S}=-\iiint\nabla\times\mathbf{F}\,dV\).

Green's theorem in 3D (1st identity): \(\iiint_V\phi\nabla^2\psi\,dV=\oiint_S\phi\dfrac{\partial\psi}{\partial n}\,dS-\iiint_V\nabla\phi\cdot\nabla\psi\,dV\).

Green's 2nd identity: \(\iiint_V(\phi\nabla^2\psi-\psi\nabla^2\phi)\,dV=\oiint_S\left(\phi\dfrac{\partial\psi}{\partial n}-\psi\dfrac{\partial\phi}{\partial n}\right)dS\).

§9 · Conservative Fields & Scalar Potential

Conservative Field — Equivalences PYQ★★★

The following are equivalent for a smooth \(\mathbf{F}\) on a simply connected domain:

\(\mathbf{F}=\nabla\phi\) for some scalar \(\phi\) (potential).
\(\nabla\times\mathbf{F}=\mathbf{0}\) (irrotational).
\(\oint_C\mathbf{F}\cdot d\mathbf{r}=0\) for every closed curve \(C\).
\(\int_A^B\mathbf{F}\cdot d\mathbf{r}\) is path-independent.

Finding Scalar Potential

If \(\mathbf{F}=\nabla\phi\), then \(\phi_x=F_1,\;\phi_y=F_2,\;\phi_z=F_3\). Integrate \(\phi_x\) w.r.t. \(x\), use \(\phi_y\) to find \(g(y,z)\), then \(\phi_z\) for the rest.

§10 · Curvilinear Coordinates

Cylindrical Coordinates \((r,\theta,z)\) PYQ★★

\(x=r\cos\theta,\;y=r\sin\theta,\;z=z\). Scale factors: \(h_r=1,\;h_\theta=r,\;h_z=1\).

\(\nabla\phi=\dfrac{\partial\phi}{\partial r}\hat r+\dfrac{1}{r}\dfrac{\partial\phi}{\partial\theta}\hat\theta+\dfrac{\partial\phi}{\partial z}\hat z\)

\(\nabla\cdot\mathbf{F}=\dfrac{1}{r}\dfrac{\partial(rF_r)}{\partial r}+\dfrac{1}{r}\dfrac{\partial F_\theta}{\partial\theta}+\dfrac{\partial F_z}{\partial z}\)

\(\nabla^2\phi=\dfrac{1}{r}\dfrac{\partial}{\partial r}\!\left(r\dfrac{\partial\phi}{\partial r}\right)+\dfrac{1}{r^2}\dfrac{\partial^2\phi}{\partial\theta^2}+\dfrac{\partial^2\phi}{\partial z^2}\)

Spherical Coordinates \((\rho,\phi,\theta)\) PYQ★★★

\(x=\rho\sin\phi\cos\theta,\;y=\rho\sin\phi\sin\theta,\;z=\rho\cos\phi\). Scale factors: \(h_\rho=1,\;h_\phi=\rho,\;h_\theta=\rho\sin\phi\).

\(\nabla^2\phi=\dfrac{1}{\rho^2}\dfrac{\partial}{\partial\rho}\!\left(\rho^2\dfrac{\partial\phi}{\partial\rho}\right)+\dfrac{1}{\rho^2\sin\phi}\dfrac{\partial}{\partial\phi}\!\left(\sin\phi\dfrac{\partial\phi}{\partial\phi}\right)+\dfrac{1}{\rho^2\sin^2\phi}\dfrac{\partial^2\phi}{\partial\theta^2}\)

Volume element: \(dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\).

Quick Application Guide

Flux out of closed surface → Gauss's theorem → volume integral.
Line integral around closed loop → Stokes' → surface integral.
Plane closed loop → Green's theorem.
Check \(\nabla\times\mathbf{F}=0\) before claiming conservative.
Simply connected domain crucial — \(\mathbf{F}=\dfrac{-y\,\hat i+x\,\hat j}{x^2+y^2}\) has curl 0 but is not conservative on \(\mathbb{R}^2\setminus\{0\}\).