Vector Integral Theorems & Coordinates

Vector integrals form the core mathematical foundation for evaluating work done along curves, mass distributions across solid volumes, and flux of gravitational and electric fields.

1. Line Integrals and Path Independence

The line integral of a vector field $\mathbf{F}$ along a directed smooth curve $C$ parameterized by $\mathbf{r}(t)$ for $t \in [a, b]$ is: $$W = \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} dt$$ For a conservative force field $\mathbf{F} = -\boldsymbol{\nabla}U$, the line integral depends strictly on the endpoints $A$ and $B$: $$\int_A^B \mathbf{F} \cdot d\mathbf{r} = -\int_A^B dU = -(U(B) - U(A)) = U(A) - U(B)$$ Consequently, the circulation around any closed loop vanishes identically: $$\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$$

2. Gauss’s Divergence Theorem

Gauss's divergence theorem establishes that the total outward flux of a continuously differentiable vector field $\mathbf{V}$ across a closed boundary surface $S = \partial V$ equals the volume integral of its divergence: $$\oint_S \mathbf{V} \cdot d\mathbf{A} = \iiint_V (\boldsymbol{\nabla} \cdot \mathbf{V}) dV$$ Gravitational Application: For the Newtonian gravitational field $\mathbf{g} = -G \frac{M}{r^2} \hat{\mathbf{r}}$, the flux across any closed surface bounding mass $M_{\text{enc}}$ is: $$\oint_S \mathbf{g} \cdot d\mathbf{A} = -4\pi G M_{\text{enc}} = -4\pi G \iiint_V \rho(\mathbf{r}) dV$$ In differential form, this produces the field equation $\boldsymbol{\nabla} \cdot \mathbf{g} = -4\pi G \rho$.

3. Stokes’ Curl Theorem

Stokes' theorem transforms the surface integral of the curl over an open orientable surface $S$ into the line integral around its bounding closed contour $C = \partial S$: $$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\boldsymbol{\nabla} \times \mathbf{F}) \cdot d\mathbf{A}$$ If $\boldsymbol{\nabla} \times \mathbf{F} = \mathbf{0}$ throughout $S$, the contour integral is guaranteed to be zero for any closed path.

Green's Theorem is the two-dimensional planar specialization of Stokes' Theorem, establishing an equivalence between a line integral around a simple closed curve and a double integral over the bounded plane region.

1. Formal Statement of Green’s Theorem

Let $C$ be a positively oriented (counterclockwise), piecewise-smooth, simple closed curve in the $xy$-plane, and let $D$ be the region bounded by $C$. If $L(x, y)$ and $M(x, y)$ have continuous partial derivatives on an open region containing $D$, then: $$\oint_C (L\, dx + M\, dy) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA$$

2. Planar Area Computation via Contour Integration

By strategically choosing functions $L$ and $M$ such that $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$:

• Case 1: $L = 0, M = x \implies \text{Area}(D) = \oint_C x\, dy$
• Case 2: $L = -y, M = 0 \implies \text{Area}(D) = -\oint_C y\, dx$
• Case 3 (Symmetric Form): $$\text{Area}(D) = \frac{1}{2} \oint_C (x\, dy - y\, dx)$$

This formula allows the exact calculation of areas bounded by parametric curves (such as ellipses, astroids, and cardioids) via simple one-dimensional boundary integrals.

For central forces (e.g., planetary orbits) and rotational motion, planar polar coordinates $(r, \theta)$ offer a vastly superior natural framework compared to Cartesian coordinates.

1. Polar Basis Vectors and Transformations

Coordinates are related by: $$x = r \cos \theta, \quad y = r \sin \theta, \quad r = \sqrt{x^2 + y^2}, \quad \theta = \arctan\left(\frac{y}{x}\right)$$ The orthonormal basis vectors are: $$\hat{\mathbf{r}} = \cos \theta \hat{\mathbf{i}} + \sin \theta \hat{\mathbf{j}}, \quad \hat{\boldsymbol{\theta}} = -\sin \theta \hat{\mathbf{i}} + \cos \theta \hat{\mathbf{j}}$$ Unlike Cartesian unit vectors, $\hat{\mathbf{r}}$ and $\hat{\boldsymbol{\theta}}$ vary with position as the angle $\theta(t)$ changes in time! Differentiating with respect to time using the chain rule: $$\frac{d\hat{\mathbf{r}}}{dt} = (-\sin\theta\dot{\theta})\hat{\mathbf{i}} + (\cos\theta\dot{\theta})\hat{\mathbf{j}} = \dot{\theta} \hat{\boldsymbol{\theta}}$$ $$\frac{d\hat{\boldsymbol{\theta}}}{dt} = (-\cos\theta\dot{\theta})\hat{\mathbf{i}} - (\sin\theta\dot{\theta})\hat{\mathbf{j}} = -\dot{\theta} \hat{\mathbf{r}}$$

2. Kinematic Velocity in Polar Coordinates

The position vector is simply $\mathbf{r} = r \hat{\mathbf{r}}$. Differentiating: $$\mathbf{v} = \frac{d\mathbf{r}}{dt} = \frac{dr}{dt} \hat{\mathbf{r}} + r \frac{d\hat{\mathbf{r}}}{dt} = \dot{r} \hat{\mathbf{r}} + r \dot{\theta} \hat{\boldsymbol{\theta}}$$

• $v_r = \dot{r}$: Radial velocity.
• $v_\theta = r\dot{\theta}$: Transverse (azimuthal) velocity.
• Speed: $v = \sqrt{\dot{r}^2 + r^2\dot{\theta}^2}$.

3. Kinematic Acceleration in Polar Coordinates

Differentiating velocity with respect to time: $$\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d}{dt}(\dot{r} \hat{\mathbf{r}} + r\dot{\theta} \hat{\boldsymbol{\theta}}) = \ddot{r}\hat{\mathbf{r}} + \dot{r}\dot{\hat{\mathbf{r}}} + \dot{r}\dot{\theta}\hat{\boldsymbol{\theta}} + r\ddot{\theta}\hat{\boldsymbol{\theta}} + r\dot{\theta}\dot{\hat{\boldsymbol{\theta}}}$$ Substituting $\dot{\hat{\mathbf{r}}} = \dot{\theta}\hat{\boldsymbol{\theta}}$ and $\dot{\hat{\boldsymbol{\theta}}} = -\dot{\theta}\hat{\mathbf{r}}$: $$\mathbf{a} = (\ddot{r} - r\dot{\theta}^2) \hat{\mathbf{r}} + (r\ddot{\theta} + 2\dot{r}\dot{\theta}) \hat{\boldsymbol{\theta}}$$

• Radial Acceleration $a_r = \ddot{r} - r\dot{\theta}^2$: Composed of linear radial acceleration $\ddot{r}$ and the inward centripetal acceleration $-r\dot{\theta}^2$.
• Transverse Acceleration $a_\theta = r\ddot{\theta} + 2\dot{r}\dot{\theta} = \frac{1}{r}\frac{d}{dt}(r^2 \dot{\theta})$: Contains the angular acceleration term $r\ddot{\theta}$ and the **Coriolis term** $2\dot{r}\dot{\theta}$.
• Kepler's Second Law: For a central force, $F_\theta = 0 \implies a_\theta = 0 \implies \frac{d}{dt}(r^2\dot{\theta}) = 0 \implies r^2\dot{\theta} = \text{const}$, proving that areal velocity is strictly constant!

Physical problems with axial symmetry (flywheels, rods) or central symmetry (gravitation, electrostatic potentials) are vastly simplified by cylindrical or spherical coordinates.

1. Cylindrical Coordinates $(r, \theta, z)$

Transformation to Cartesian coordinates: $$x = r \cos \theta, \quad y = r \sin \theta, \quad z = z$$ Orthonormal basis vectors: $$\hat{\mathbf{r}} = \cos \theta \hat{\mathbf{i}} + \sin \theta \hat{\mathbf{j}}, \quad \hat{\boldsymbol{\theta}} = -\sin \theta \hat{\mathbf{i}} + \cos \theta \hat{\mathbf{j}}, \quad \hat{\mathbf{z}} = \hat{\mathbf{k}}$$ Metric scale factors: $h_r = 1, h_\theta = r, h_z = 1$. Infinitesimal line element: $$d\mathbf{r} = dr \hat{\mathbf{r}} + r d\theta \hat{\boldsymbol{\theta}} + dz \hat{\mathbf{z}}, \quad dV = r \, dr \, d\theta \, dz$$ Differential operators in cylindrical coordinates: $$\boldsymbol{\nabla}\Phi = \frac{\partial \Phi}{\partial r} \hat{\mathbf{r}} + \frac{1}{r} \frac{\partial \Phi}{\partial \theta} \hat{\boldsymbol{\theta}} + \frac{\partial \Phi}{\partial z} \hat{\mathbf{z}}$$ $$\boldsymbol{\nabla} \cdot \mathbf{V} = \frac{1}{r}\frac{\partial(r V_r)}{\partial r} + \frac{1}{r}\frac{\partial V_\theta}{\partial \theta} + \frac{\partial V_z}{\partial z}$$ $$\nabla^2 \Phi = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \Phi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \Phi}{\partial \theta^2} + \frac{\partial^2 \Phi}{\partial z^2}$$

2. Spherical Polar Coordinates $(r, \theta, \phi)$

Transformation ($ heta$ is colatitude / polar angle, $\phi$ is azimuth): $$x = r \sin \theta \cos \phi, \quad y = r \sin \theta \sin \phi, \quad z = r \cos \theta$$ Scale factors: $h_r = 1, h_\theta = r, h_\phi = r \sin \theta$. Differential volume element: $$dV = r^2 \sin \theta \, dr \, d\theta \, d\phi$$ Differential operators in spherical coordinates: $$\boldsymbol{\nabla}\Phi = \frac{\partial \Phi}{\partial r} \hat{\mathbf{r}} + \frac{1}{r} \frac{\partial \Phi}{\partial \theta} \hat{\boldsymbol{\theta}} + \frac{1}{r \sin \theta} \frac{\partial \Phi}{\partial \phi} \hat{\boldsymbol{\phi}}$$ $$\boldsymbol{\nabla} \cdot \mathbf{V} = \frac{1}{r^2}\frac{\partial(r^2 V_r)}{\partial r} + \frac{1}{r \sin \theta}\frac{\partial(\sin \theta V_\theta)}{\partial \theta} + \frac{1}{r \sin \theta}\frac{\partial V_\phi}{\partial \phi}$$ $$\nabla^2 \Phi = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial \Phi}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \Phi}{\partial \theta}\right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \Phi}{\partial \phi^2}$$