23.2 Surface integrals of vector fields

Surface integrals of vector fields measure the flow through a surface, like fluid through a pipe or electric field lines through a charged surface. They're calculated by integrating the dot product of the vector field and surface normal vector over the surface.

This topic builds on earlier concepts of vector fields and surface integrals, applying them to real-world scenarios. It's crucial for understanding flux in physics and engineering, connecting mathematical theory to practical applications in fluid dynamics and electromagnetism.

Vector Fields and Surface Integrals

Vector Field Fundamentals

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• Vector field assigns a vector to each point in a subset of space
  • Can be 2D or 3D
  • Commonly used to model physical quantities with magnitude and direction (velocity fields, force fields, electric fields)
• Flux integral measures the total flow of a vector field through a surface
  • Calculated by integrating the dot product of the vector field and the surface normal vector over the surface
  • Represents the amount of "stuff" passing through the surface (fluid, electric field, heat)
• Surface normal vector is a unit vector perpendicular to the surface at each point
  • Points outward for a closed surface (sphere, cube)
  • Determines the orientation of the surface for flux calculations
• Dot product of two vectors $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$
  • Measures the alignment of the vectors
  • Maximum when vectors are parallel, zero when perpendicular

Calculating Surface Integrals

• To evaluate a surface integral, parametrize the surface using two variables $(u,v)$
  • $\vec{r}(u,v) = (x(u,v), y(u,v), z(u,v))$
• Compute the surface normal vector using the cross product of partial derivatives
  • $\vec{n} = \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}$
• Integrate the dot product of the vector field $\vec{F}$ and surface normal vector over the parameter domain
  • $\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F}(\vec{r}(u,v)) \cdot \left(\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}\right) \, du \, dv$
• Surface integrals can be used to find the total flux through a surface (flow rate through a pipe, electric flux through a charged surface)

Orientation and the Divergence Theorem

Surface Orientation

• Orientation of a surface determines the direction of the surface normal vector
  • Two possible orientations for a surface: inward-pointing or outward-pointing normal vectors
  • Closed surfaces (sphere, cube) have an outward orientation by convention
  • Open surfaces (part of a plane, paraboloid) can have either orientation
• Orientation affects the sign of the flux integral
  • Positive flux for outward-pointing normal vectors
  • Negative flux for inward-pointing normal vectors
• Consistency in orientation is crucial when applying the divergence theorem

Divergence Theorem

• Divergence theorem relates the flux of a vector field through a closed surface to the divergence of the vector field within the enclosed volume
  • Also known as Gauss's theorem or Ostrogradsky's theorem
• Divergence of a vector field $\vec{F} = (P,Q,R)$ is $\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$
  • Measures the net outward flow of the vector field from a point
• Divergence theorem states: $\iint_S \vec{F} \cdot d\vec{S} = \iiint_V \nabla \cdot \vec{F} \, dV$
  • Left side is the flux integral over the closed surface $S$
  • Right side is the triple integral of the divergence over the enclosed volume $V$
• Useful for converting surface integrals to volume integrals (electric flux, heat flow)

Applications

Fluid Dynamics

• Vector fields are used to model fluid flow (velocity fields, pressure gradients)
  • Velocity field $\vec{v}(x,y,z)$ gives the velocity of the fluid at each point
  • Pressure gradient $\nabla p$ points in the direction of greatest pressure change
• Flux integrals can determine the flow rate through a surface (pipe, vent, membrane)
  • $Q = \iint_S \vec{v} \cdot d\vec{S}$
  • Positive flux for outward flow, negative for inward flow
• Divergence theorem relates the net outward flow to the divergence of the velocity field
  • Incompressible fluids have zero divergence ($\nabla \cdot \vec{v} = 0$)
  • Non-zero divergence indicates sources or sinks in the flow

Electromagnetism

• Electric and magnetic fields are vector fields
  • Electric field $\vec{E}(x,y,z)$ points in the direction of the force on a positive charge
  • Magnetic field $\vec{B}(x,y,z)$ points in the direction of the force on a moving charge
• Electric flux through a surface is the surface integral of the electric field
  • $\Phi_E = \iint_S \vec{E} \cdot d\vec{S}$
  • Measures the total number of field lines passing through the surface
• Magnetic flux through a surface is the surface integral of the magnetic field
  • $\Phi_B = \iint_S \vec{B} \cdot d\vec{S}$
  • Relates to the induced electromotive force (EMF) in a changing magnetic field (Faraday's law)
• Divergence theorem applies to electric fields (Gauss's law)
  • Net electric flux through a closed surface is proportional to the enclosed charge
  • $\iint_S \vec{E} \cdot d\vec{S} = \frac{Q}{\epsilon_0}$, where $Q$ is the enclosed charge and $\epsilon_0$ is the permittivity of free space