🧮Physical Sciences Math Tools Unit 4 – Vector Function & Field Integration

Key Concepts

• Vector functions map real numbers to vectors in 2D or 3D space
• Vector fields assign a vector to each point in a subset of space
• Line integrals calculate the work done by a vector field along a curve
  • Fundamental theorem of line integrals relates line integrals to potential functions
• Surface integrals measure the flux of a vector field through a surface
• Divergence measures how much a vector field spreads out from a point
  • Divergence theorem relates surface integrals to volume integrals
• Curl measures the rotation of a vector field around a point
  • Stokes' theorem relates line integrals to surface integrals of curl
• Applications include fluid dynamics, electromagnetism, and gravity

Vector Functions Basics

• A vector function $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ maps a scalar parameter $t$ to a vector
• The domain of a vector function is a subset of real numbers
• The range of a vector function is a subset of 2D or 3D space
• Continuity and differentiability of vector functions depend on their component functions
  • If $f(t)$, $g(t)$, and $h(t)$ are continuous (differentiable), then $\vec{r}(t)$ is continuous (differentiable)
• The derivative of a vector function is another vector function $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$
• Integrals of vector functions are evaluated component-wise
  • $\int \vec{r}(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle$

Vector Fields and Their Properties

• A vector field $\vec{F}(x, y, z) = \langle P(x, y, z), Q(x, y, z), R(x, y, z) \rangle$ assigns a vector to each point in a subset of 3D space
• Vector fields can be visualized using arrows or streamlines
  • Arrow plots show the direction and magnitude of the field at each point
  • Streamlines are curves tangent to the field at each point
• Conservative vector fields have a potential function $\phi$ such that $\vec{F} = \nabla \phi$
  • The work done by a conservative field is path-independent
• Irrotational vector fields have zero curl ($\nabla \times \vec{F} = \vec{0}$)
  • Conservative fields are always irrotational, but not all irrotational fields are conservative
• Solenoidal vector fields have zero divergence ($\nabla \cdot \vec{F} = 0$)
  • Incompressible fluid flows are examples of solenoidal fields

Line Integrals

• A line integral $\int_C \vec{F} \cdot d\vec{r}$ measures the work done by a vector field $\vec{F}$ along a curve $C$
• Parametric equations $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$ describe the curve $C$
• The line integral is evaluated using the dot product and a parameter $t$
  • $\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt$
• For conservative fields, the fundamental theorem of line integrals states that $\int_C \vec{F} \cdot d\vec{r} = \phi(\vec{r}(b)) - \phi(\vec{r}(a))$
  • The line integral depends only on the endpoints of the curve, not the path taken
• Line integrals have applications in work, circulation, and flux calculations

Surface Integrals

• A surface integral $\iint_S \vec{F} \cdot d\vec{S}$ measures the flux of a vector field $\vec{F}$ through a surface $S$
• Parametric equations $\vec{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle$ describe the surface $S$
• The surface integral is evaluated using the dot product and parameters $u$ and $v$
  • $\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F}(\vec{r}(u, v)) \cdot (\vec{r}_u \times \vec{r}_v) du dv$
• The orientation of the surface affects the sign of the flux
  • Outward-pointing normal vectors give positive flux, inward-pointing give negative flux
• Surface integrals have applications in fluid dynamics, electromagnetism, and heat transfer

Flux and Divergence

• The flux of a vector field $\vec{F}$ through a closed surface $S$ is $\Phi = \oiint_S \vec{F} \cdot d\vec{S}$
• The divergence of a vector field $\vec{F}$ at a point is $\nabla \cdot \vec{F} = \lim_{V \to 0} \frac{1}{V} \oiint_S \vec{F} \cdot d\vec{S}$
  • Divergence measures how much the field spreads out from a point
• The divergence theorem (Gauss' theorem) states that $\oiint_S \vec{F} \cdot d\vec{S} = \iiint_V \nabla \cdot \vec{F} dV$
  • The flux through a closed surface equals the volume integral of the divergence
• Positive divergence indicates a source, negative divergence indicates a sink
• The divergence of a solenoidal field is zero everywhere

Curl and Stokes' Theorem

• The curl of a vector field $\vec{F}$ at a point is $\nabla \times \vec{F} = \lim_{A \to 0} \frac{1}{A} \oint_C \vec{F} \cdot d\vec{r}$
  • Curl measures the rotation of the field around a point
• Stokes' theorem states that $\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$
  • The line integral of a vector field around a closed curve equals the surface integral of its curl
• The curl of a conservative (irrotational) field is zero everywhere
• Stokes' theorem has applications in fluid dynamics and electromagnetism
  • Relates circulation to vorticity and magnetic fields to electric currents

Applications in Physics

• Fluid dynamics: Velocity fields of fluids are vector fields
  • Divergence measures compressibility, curl measures vorticity
  • Navier-Stokes equations involve divergence and curl of velocity and pressure fields
• Electromagnetism: Electric and magnetic fields are vector fields
  • Gauss' law relates electric flux to charge density using divergence
  • Faraday's law relates magnetic flux to induced electric fields using curl
  • Ampère's law relates magnetic fields to electric currents using curl
• Gravity: Gravitational fields are conservative vector fields
  • Gravitational potential energy is a scalar potential function
  • Work done by gravity is path-independent, depends only on start and end points
• Heat transfer: Temperature gradients are vector fields
  • Fourier's law relates heat flux to temperature gradient using divergence
  • Conservation of energy leads to the heat equation involving divergence