Source: Reddit - I made a little graphic for the Fundamental Theorems of Calc III

1 Unified Framework: Generalized Stokes’ Theorem

All fundamental theorems of vector calculus are specific dimensional cases of the Generalized Stokes’ Theorem:

$\int_{\Omega} d\omega = \int_{\partial \Omega} \omega$

Where $\Omega$ represents the interior region, $\partial \Omega$ denotes its boundary, and $d\omega$ represents the exterior derivative. The core principle is that the integration of a derivative over a domain equals the integration of the original form over its boundary.

1D: Fundamental Theorem of Calculus

• Formula: $\int_a^b f'(x) \, dx = f(b) - f(a)$
• Domain: 1D interval $[a, b]$ with 0D boundary endpoints $a$ and $b$.

2D: Green’s Theorem

• Formula: $\iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dxdy = \oint_C (P\,dx + Q\,dy)$
• Domain: 2D planar region $D$ bounded by a 1D closed curve $C$.

3D Surfaces: Stokes’ Theorem (Curl Theorem)

• Formula: $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$
• Interpretation: The surface integral of the curl of a vector field $\mathbf{F}$ over an open surface $S$ equals the line integral of $\mathbf{F}$ along the boundary curve $\partial S$.

3D Volumes: Gauss’s Theorem (Divergence Theorem)

• Formula: $\iiint_V (\nabla \cdot \mathbf{F}) \, dV = \oiint_{\partial V} \mathbf{F} \cdot d\mathbf{S}$
• Interpretation: The volume integral of the divergence of a vector field $\mathbf{F}$ over a solid $V$ equals the net flux of $\mathbf{F}$ passing through the closed bounding surface $\partial V$.

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2 Operational Rules of the Del ($\nabla$) Operator

The del operator $\nabla$ functions simultaneously as a vector and a differential operator. When applied to products, it follows the product rule (Leibniz rule) rather than the chain rule.

Vector Identity: $\nabla \times (\varphi \mathbf{A})$

$\nabla \times (\varphi \mathbf{A}) = \varphi (\nabla \times \mathbf{A}) + (\nabla \varphi) \times \mathbf{A}$

Derivation and Structural Mechanics:
By applying the product rule, $\nabla$ differentiates $\varphi$ and $\mathbf{A}$ sequentially:

1. When differentiating $\mathbf{A}$, the scalar field $\varphi$ acts as a constant coefficient and is factored out, yielding $\varphi (\nabla \times \mathbf{A})$.
2. When differentiating $\varphi$, $\nabla$ operates directly on the scalar to produce the gradient vector $\nabla \varphi$. The underlying cross product structure ($\times$) with the remaining vector field $\mathbf{A}$ must be preserved, yielding $(\nabla \varphi) \times \mathbf{A}$.

This can be verified by expanding the $x$-component using partial derivatives:
$\left[\nabla \times (\varphi \mathbf{A})\right]_x = \frac{\partial}{\partial y}(\varphi A_z) - \frac{\partial}{\partial z}(\varphi A_y)$
Applying the standard product rule:
$= \left( \varphi \frac{\partial A_z}{\partial y} + A_z \frac{\partial \varphi}{\partial y} \right) - \left( \varphi \frac{\partial A_y}{\partial z} + A_y \frac{\partial \varphi}{\partial z} \right)$
Rearranging the terms:
$= \varphi \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) + \left( \frac{\partial \varphi}{\partial y}A_z - \frac{\partial \varphi}{\partial z}A_y \right) = \left[ \varphi (\nabla \times \mathbf{A}) + (\nabla \varphi) \times \mathbf{A} \right]_x$

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3 Summary

Theorem	Interior ($\Omega$)	Boundary ($\partial \Omega$)	Operator
Fundamental Theorem	1D Segment	0D Endpoints	$d/dx$
Stokes’ Theorem	2D Surface	1D Curve	Curl ($\nabla \times$)
Divergence Theorem	3D Volume	2D Closed Surface	Divergence ($\nabla \cdot$)