Vector Calculus: Functions and Fields

Vector Calculus Overview

• Vector calculus combines differentiation, integration, and vector functions.
• It provides mathematical tools to study object behavior in coordinate systems.
• It can be used to determine the rate of change of a function describing an object moving in 3D space.
• Example: The position of a plane relative to an observer can be described by a time-dependent vector (x(t), y(t), z(t)), where the rate of change of this position vector gives the velocity, and the rate of change of the velocity gives the acceleration.
• Scalar and vector fields are important concepts.
• Scalar Field: Function that gives a scalar value at each point in space.
• Example of scalar field: Temperature at each point in a room; given a position, the function outputs a scalar.
• Vector Field: Associates a vector (velocity, speed, direction) with each point in space.
• Example of vector field: Speed and direction of water flowing down a drain.

Differentiation of Vector Functions

• The derivative indicates rate of change of a function with respect to its arguments.
• The position of a car and its velocity can be described with vectors.
• The position of the car is a vector function with a scalar argument (time).
• A vector function a with scalar argument u in 3D Cartesian space can be expressed as a = ax(u)i + ay(u)j + az(u)k, where i, j, and k are unit vectors in the x-, y-, and z-directions, and ax(u), ay(u), and az(u) are the components in each direction.
• The derivative of a vector a(u) can be defined as: da/du = lim (Δu→0) [a(u + Δu) - a(u)] / Δu
• The derivative of a vector function is also a vector function.
• They aren't necessarily parallel.
• da/du = (dax/du)i + (day/du)j + (daz/du)k.
• Each component of vector function a(u) can be differentiated separately.

Rules of Differentiation of Vector Functions

• Just as for scalar functions, rules of differentiation allow us to avoid directly using the definition.
• a and b are differentiable vector functions and Φ is a differentiable scalar function.
• d(Φa)/du = Φ(da/du) + (dΦ/du)a
• d(a⋅b)/du = a⋅(db/du) + (da/du)⋅b
• d(a×b)/du = a×(db/du) + (da/du)×b

Vector Functions with Multiple Scalar Arguments

• Partial derivatives express the rate of change of a function with respect to a single variable.
• The idea of partial derivatives can be extended to vector functions that depend on more than one variable.
• If a(u1, u2, ..., un) is a vector function with scalar arguments u1, ..., un, then to find ∂a/∂ui, treat all variables uj where j ≠ i as constant and differentiate a as we vary only ui.
• Given that the arguments u₁, u₂, ..., un are functions of some variables v₁, v₂, ..., vm, the chain rule becomes: (da/dvᵢ) = (∂a/∂u₁) (∂u₁/∂vᵢ) + (∂a/∂u₂) (∂u₂/∂vᵢ) + ... + (∂a/∂un) (∂un/∂vᵢ)

Integration of Vectors

• Integration of vector functions is analogous to integrating functions of a single variable.
• Given the vector function a(u) = dA(u)/du, the indefinite integral is ∫a(u) du = A(u) + b, where "b" is an arbitrary constant vector.
• Definite integral: ∫[u1, u2] a(u) du = A(u2) - A(u1)
• The antiderivative of a vector function is a vector function.
• The constant of integration is a vector constant.

Integration along Paths

• Integrals can be performed along arbitrary paths.
• Work performed when applying a force along a path is an example.
• In the simplest case, work W = Fr.
• W is a scalar
• F and r are vectors
• Work performed by a force F as a particle travels along path C: W = ∫C F ⋅ dr.
• The component of the force parallel to the line tangent to the curve contributes to the work done.
• r(t) = (x(t), y(t), z(t)) parameterizes the path.
• dr(t) = (dx, dy, dz) = (dx/dt, dy/dt, dz/dt) dt.

Integration over Surfaces

• Double integration can be extended to arbitrary surfaces.
• A single-parameter describes movement along a fixed curve.
• Two free variables describe a surface.
• Example: r(u, v) = r₀ + ua + vb.
• r₀ is a fixed point on the surface.
• It anchors the surface in space.
• Linear combinations of vectors a and b span the surface.
• A small surface area generated by a and b will be a parallelogram, dA = |(∂r/∂u) x (∂r/∂v)| dudv.

Scalar and Vector Fields

• Scalar Field: A function that assigns a scalar value to each point in a space.
• Example: Temperature(x, y), assigning temperature to each point in a 2D space.
• Vector Field: Each point in a region is associated with a movement or force in a given direction with some strength.
• Example: Velocity(x,y,z,t)
• To visualize a vector field, arrows are placed to indicate the direction and strength at each point.
• Color coding can also be used.
• Stream plots can be used to show how the vector field changes, with little arrows on field lines.

Vector Operations

• The vector operator ∇ (nabla/del) is used in applications of derivatives of vector fields.
• In Cartesian coordinates, ∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k.
• ∇² = Δ = (∂²/∂x²) + (∂²/∂y²) + (∂²/∂z²) (Delta/Laplace operator)

Gradient of a Scalar Field

• One of the most important applications of the nabla operator is determining the rate of change of a scalar field Φ in a given direction, called the directional derivative.
• dΦ(s)/ds = a ⋅ ∇Φ
• ∇Φ is the gradient of a scalar field; it describes the direction of steepest ascent.

Divergence of a Vector Field

• The scalar product of the nabla operator with a vector field is called the divergence of a vector field.
• ∇⋅V = div V
• The divergence measures the flux of a vector field at a point.
• It can be interpreted as how much the field acts like a source or drain at the point.
• If the total volume of water is unchanged between the entrance and exit to a pipe, then there are no sources.
• The divergence of V =O

Curl of a Vector Field

• The cross product of the nabla operator and a vector field.
• ∇ x V = curl V
• The curl describes the "whirliness" of a vector field.
• Represents the angular velocity of water around a point.