2.6. Divergence Theorem and Stokes' Theorem

Divergence Theorem

The Divergence Theorem relates the flux of a vector field through a closed surface to the divergence of the field within the enclosed volume. Mathematically:\[\iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV,\] where:

  •  \( \mathbf{F} \): Vector field (e.g., flow velocity or electric field).
  •  \( \partial V \): Closed surface enclosing the volume \(V\).
  •  The divergence of a vector field \(\mathbf{F} \), denoted as \(\nabla \cdot \mathbf{F}\), is a scalar quantity that measures the rate at which the vector field spreads out or converges at a point. It is defined as: \[ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}, \]
  • Proof of Divergence Theorem 
    • Step 1: Verification for a Cuboid. Consider a cuboidal volume: \[ V = \{ (x, y, z) : 0 < x < a, 0 < y < b, 0 < z < c \}. \] The integral over \( V \) is: \[ \iiint_V \nabla \cdot \mathbf{F} \, dV = \iiint_V \left( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \right) dx \, dy \, dz. \] Using the Fundamental Theorem of Calculus for each dimension: - For \( F_x \), integrate over \( x \): \[ \int_0^a \frac{\partial F_x}{\partial x} \, dx = F_x(a, y, z) - F_x(0, y, z). \] - For \( F_y \), integrate over \( y \): \[ \int_0^b \frac{\partial F_y}{\partial y} \, dy = F_y(x, b, z) - F_y(x, 0, z). \] - For \( F_z \), integrate over \( z \): \[ \int_0^c \frac{\partial F_z}{\partial z} \, dz = F_z(x, y, c) - F_z(x, y, 0). \] Summing these contributions gives: \[ \iiint_V \nabla \cdot \mathbf{F} \, dV = \underbrace{\iint F_x(a, y, z) - F_x(0, y, z) \, dy \, dz}_{A_x} + A_y+A_z\] where \[ A_y +A_z=\iint F_y(x, b, z) - F_y(x, 0, z) \, dx \, dz + \iint F_z(x, y, c) - F_z(x, y, 0) \, dx \, dy. \] This matches the surface integral: \[ \iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS, \] validating the theorem for a cuboid.
    • Step 2: Union of Two Cuboids. Let \( V_1 \) and \( V_2 \) be two cuboids, and let \( V = V_1 \cup V_2 \). The divergence theorem holds for each volume individually: \[ \iiint_{V_1} \nabla \cdot \mathbf{F} \, dV = \iint_{\partial V_1} \mathbf{F} \cdot \mathbf{n} \, dS, \quad \iiint_{V_2} \nabla \cdot \mathbf{F} \, dV = \iint_{\partial V_2} \mathbf{F} \cdot \mathbf{n} \, dS. \] At the common interface \( \partial V_1 \cap \partial V_2 \), the outward normal vectors \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) are opposite. Therefore, the flux through the common surface cancels: \[ \iint_{\partial V_1 \cap \partial V_2} \mathbf{F} \cdot \mathbf{n}_1 \, dS + \iint_{\partial V_2 \cap \partial V_1} \mathbf{F} \cdot \mathbf{n}_2 \, dS = 0. \] Thus, for the union \( V \): \[ \iiint_V \nabla \cdot \mathbf{F} \, dV = \iiint_{V_1} \nabla \cdot \mathbf{F} \, dV + \iiint_{V_2} \nabla \cdot \mathbf{F} \, dV = \iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS. \]
    • Step 3: Generalization to Arbitrary Volumes: Any arbitrary volume \(V\) can be approximated as a union of cuboidal subvolumes. 
  • Example 1: Consider a section of a blood vessel as a closed surface \( \partial V \) enclosing the blood volume \( V \). Let \( \mathbf{F} \) represent the blood velocity field, where \( \mathbf{F} \cdot \mathbf{n} \) represents the blood flow rate through the vessel walls. The net flow rate through the vessel walls is given by: \[ \iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS. \] Using the Divergence Theorem: \[ \iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV. \] Here:  \( \nabla \cdot \mathbf{F} \) measures the divergence of the velocity field, which represents the rate at which blood is expanding or compressing within the volume \( V \). - If \( \nabla \cdot \mathbf{F} > 0 \), there is a net outward flow (e.g., due to an aneurysm). - If \( \nabla \cdot \mathbf{F} < 0 \), there is a net inward flow (e.g., due to vessel constriction). 
  • Example 2: Measuring Blood Supply to an Organ. Suppose the divergence \( \nabla \cdot \mathbf{F} \) is measured within a region of the brain or kidney. Integrating this divergence over the volume of interest provides the net blood supply to the organ: \[ \iiint_V (\nabla \cdot \mathbf{F}) \, dV. \] This is critical in assessing conditions like ischemia (reduced blood supply) or hyperperfusion (excessive blood flow). 

Stokes' Theorem

Stokes' Theorem relates the circulation of a vector field, \(\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}\), around a closed curve to the surface integral of the curl of the field, \(\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dA\), over a surface \(S\) bounded by that curve. That is, 

\[\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dA,\] where:

    •  \( \mathbf{F} \): Vector field (e.g., velocity field of blood flow).
    • \(\nabla \times \mathbf{F}\): Curl of the vector field, defined as: \[ \nabla \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}, \]
    •  \( \partial S \): Boundary curve of the surface \(S\).
    •  \( \mathbf{n} \): Unit normal vector to the surface \(S\).
  • Proof. 
    • Step 1: Proof for the Special Case of a Rectangular Surface. Let \( S = \{ (x, y) : 0 < x < a, 0 < y < b \} \), a rectangular surface in the \( xy \)-plane. For this surface, the left-hand side of Stokes' Theorem becomes: \[ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dx \, dy = \iint_S \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) dx \, dy. \] By integrating: \[ \int_0^b \int_0^a \frac{\partial F_2}{\partial x} dx \, dy = \int_0^b \left[ F_2(a, y) - F_2(0, y) \right] dy, \] and \[ \int_0^a \int_0^b \frac{\partial F_1}{\partial y} dy \, dx = \int_0^a \left[ F_1(x, b) - F_1(x, 0) \right] dx. \] The surface integral simplifies to: \[ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dx \, dy = \int_0^b \left[ F_2(a, y) - F_2(0, y) \right] dy - \int_0^a \left[ F_1(x, b) - F_1(x, 0) \right] dx. \] For the boundary line integral, consider \( \partial S \), the rectangular contour traversed counter-clockwise: \[ \oint_{\partial S} \mathbf{F} \cdot d\boldsymbol{\ell} = \int_{(0, 0) \to (a, 0)} \mathbf{F} \cdot d\boldsymbol{\ell} + \int_{(a, 0) \to (a, b)} \mathbf{F} \cdot d\boldsymbol{\ell} + \int_{(a, b) \to (0, b)} \mathbf{F} \cdot d\boldsymbol{\ell} + \int_{(0, b) \to (0, 0)} \mathbf{F} \cdot d\boldsymbol{\ell}. \] Summing these contributions verifies Stokes' Theorem for the rectangular surface. ---
    •  Step 2: Proof for the Union of Two Regions. Let \( S = S_1 \cup S_2 \), where \( S_1 \) and \( S_2 \) are two adjacent regions with a shared boundary. Using the additivity of the surface integral: \[ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \iint_{S_1} (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS + \iint_{S_2} (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS. \] For the boundary integral, the shared boundary is traversed in opposite directions for \( \partial S_1 \) and \( \partial S_2 \), so the contributions cancel: \[ \oint_{\partial S_1} \mathbf{F} \cdot d\boldsymbol{\ell} + \oint_{\partial S_2} \mathbf{F} \cdot d\boldsymbol{\ell} = \oint_{\partial S} \mathbf{F} \cdot d\boldsymbol{\ell}. \] Thus, the theorem holds for \( S = S_1 \cup S_2 \). --
    • Step 3: Generalization to Arbitrary Surfaces. Any arbitrary surface \( S \) can be approximated as a union of small planar elements. By applying the above steps to each small element and summing, Stokes' Theorem is proven for arbitrary surfaces.
  • Example 1. Electromagnetic Field Analysis. In electromagnetism, Stokes' Theorem is used to relate the electric field \(\mathbf{E}\) and the magnetic field \(\mathbf{B}\) in Maxwell's equations. For instance: \[ \oint_{\partial S} \mathbf{E} \cdot d\mathbf{r} = -\iint_S \frac{\partial \mathbf{B}}{\partial t} \cdot \mathbf{n} \, dA, \] which describes Faraday's Law of electromagnetic induction. This law is fundamental in designing transformers, electric motors, and generators.
  • Example 2. Fluid Flow and Circulation. In fluid dynamics, Stokes' Theorem is used to calculate the circulation of a velocity field \(\mathbf{v}\) around a closed path: \[ \oint_{\partial S} \mathbf{v} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{v}) \cdot \mathbf{n} \, dA. \] This is particularly useful for: - Measuring vorticity in turbulent blood flow. - Understanding circulation patterns in cardiovascular systems, such as near aneurysms or heart valves. 
  • Example 3. Medical Imaging and MRI. Stokes' Theorem is indirectly applied in Magnetic Resonance Imaging (MRI) to understand the relationship between changing magnetic fields and induced electric currents. Specifically, it helps model the spatial relationship between the magnetic gradient fields and the induced signal. 
  • Example 4. Magnetic Resonance-Based Blood Flow Analysis. Stokes' Theorem helps in determining circulation in blood flow using magnetic resonance imaging (MRI). The theorem relates the circulation of blood flow velocity along a closed path to the curl (vorticity) of the blood flow over a surface. This is useful in detecting abnormal flow patterns in arteries and veins. 
  • Example 5. Analysis of Eddy Currents in Conductors. Stokes' Theorem is used to analyze eddy currents in electromagnetic systems, such as in MRI or other diagnostic devices, by relating the circulation of induced currents to the curl of the magnetic field.  


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2.1 Vectors

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2.5. Vector Fields and Line integral

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