Vector Calculus: Differentiation and Integration

• Unit 4 focuses on vector calculus, combining differentiation, integration, and vector functions.

Study Goals

• Differentiate and integrate vector functions.
• Differentiate along an arbitrary line.
• Integrate over an arbitrary surface.
• Understand scalar and vector fields and their visualization.
• Use and interpret vector operators on scalar and vector fields.

Introduction to Vector Calculus

• Vector calculus extends mathematical tools to study object behavior in arbitrary coordinate systems.
• It is useful for determining rates of change of functions describing objects moving in 3D space.
• Example: tracking a plane's position (x(t), y(t), z(t)), velocity (rate of change of position), and acceleration (rate of change of velocity).
• Scalar fields: functions that assign a scalar value to each point in space (e.g., temperature in a room).
• Vector fields: assign a vector (speed and direction) to each point in space (e.g., water flow down a drain).

Differentiation of Vector Functions

• The derivative is the rate of change of a function with respect to its arguments.
• Motion along a straight road is the rate of change in the distance traveled.
• Vectors can describe position and velocity.
• A vector function à = à(u) depends on a scalar argument u.
• In 3D Cartesian space: à = ax(u)i + ay(u)j + az(u)k, where i, j, k are unit vectors.
• Components ax(u), ay(u), and az(u) are scalar functions.
• The derivative of a vector à(u) is defined as da/du = lim (Δu→0) [à(u + Δu) - à(u)] / Δu .
• The derivative of a vector function is also a vector.
• Vectors are not necessarily parallel and can point in different directions.
• In Cartesian coordinates: da/du = (dax/du)i + (day/du)j + (daz/du)k
• Each component of the vector function à(u) can be differentiated separately.
• A small change in the argument u leads to a small change in the vector.

Example of Vector Differentiation

• Position: x(t) = t²i + 3tj + tk.
• Velocity: v(t) = dx(t)/dt = 2ti + 3j + k.
• At t=1, velocity v(1) = 2i + 3j + k.
• Speed is the magnitude of velocity: |v(1)| = √(2² + 3² + 1²) = √14.

Rules of Differentiation

• Rules simplify differentiation, avoiding the definition.
• Assume differentiable vector functions à, b and scalar function Φ.
• d(Φà)/du = (dΦ/du)à + Φ(dà/du).
• d(à·b)/du = (dà/du)·b + à·(db/du).
• d(à×b)/du = (dà/du)×b + à×(db/du).

Example: Particle Circling

• Particle circling a center with constant speed and radius.
• For any time t, the velocity vector is perpendicular to the position vector.
• Position function: r(t.)
• Distance from center is constant.
• Constant magnitude: |v(t)| = v².
• r(t)·r(t) = r² indicates 0=2(r(t)dotv(t)) implying perpendicularity

Vector Functions with Multiple Scalar Arguments

• Partial derivatives express the rate of change of a function w.r.t. a single variable.
• Vector function à(u₁, u₂, ..., un) with scalar arguments u₁, ..., un.
• To find ∂à/∂ui, treat all variables uj (j≠i) as constant and differentiate à as we vary ui.
• Chain rule: dа/dvᵢ = (∂a/∂u₁) * (∂u₁/∂vᵢ) + (∂a/∂u₂) * (∂u₂/∂vᵢ) + ... + (∂a/∂un) * (∂un/∂vᵢ).

Integration of Vectors

• Integration of vector functions is analogous to integration of scalar functions.
• a = dA(u)/du, the integral is ∫a(u)du = A(u) + b, where b is an arbitrary constant vector.
• ∫[u1 to u2] a(u)du = A(u2) - A(u1) is the definite integral.
• The antiderivative of a scalar function is a scalar function.
• The antiderivative of a vector function is a vector function, and its constant of integration is a vector constant.

Integration Along Paths

• Integrals are performed along arbitrary paths.
• Work performed applying a force along a path: W = ∫C F · dr
• Only the component of the force || to the curve contributes to the work.
• Path is parameterized by r(t) = (x(t), y(t), z(t)).
• Differential: dr = (dx, dy, dz) = (dx/dt, dy/dt, dz/dt)dt.

Integration Over Surfaces

• Extend double integration to arbitrary surfaces.
• Single free parameter describes movement along a fixed curve.
• Two free variables needed to parametrize a surface.
• r(u, v) = r0 + ua + vb, is a parameterization
• r0 is a fixed point.
• Linear combinations of vectors à and b span the surface.
• Cartesian coordinates, vectors i and j are orthogonal, creating a rectangular area.
• dA = |(∂r/∂u) × (∂r/∂v)| dudv, the small surface area generated by a and b will be a parallelogram.

Scalar and Vector Fields

• Scalar field Φ(x, y) assigns a scalar value to each point in a 2D space.
• Example: temperature in a room with a heat source (radiator).
• Scalar field is a function that depends on both position and time (x, y, z, t).
• Vector field assigns a movement or force in a given direction with some strength at each point in a region.
• Intuition: the direction and speed of water change with position.
• V = V(x, y, z, t) assigns a vector (velocity of water) to each position (x, y, z) at a given time t.
• Generalize scalar and vector fields to arbitrarily many variables V = V(X1, X2, ..., Xn).
• Vector fields are more complex to visualize compared to scalar fields.
• Add information about strength and direction at each point.
• Place little arrows on a regular grid.
• Arrows and their lengths indicate the direction and strength at each point.

Visualization of Vector Field

• Color info highlights the strength of a vector field.
• Stream plot: lines show how the vector field changes as a function of position, with particles tracing the field.
• Can vary the density or the color of the stream lines to indicate the strength of the vector field.

Vector Operations

• The vector operator ∇ (nabla or del) is used in derivatives of vector fields.
• In Cartesian coordinates: ∇ = i(∂/∂x) + j(∂/∂y) + k(∂/∂z)
• Second partial derivatives are found by repeated application of ∇: Δ = ∇·∇ = (∂²/∂x²) + (∂²/∂y²) + (∂²/∂z²).
• The resulting operator is the Delta or Laplace operator.

Gradient of a Scalar Field

• The nabla operator is used to determine the rate of change of a scalar field Φ in a given direction.
• Called the directional derivative.
• Starting at point P(x₀, y₀, z₀), a small distance is moved along the line g(s, à) = x + să, given direction à.
• The field at the new point.
• The rate of change of Φ in the direction of à (directional derivative): Vₐ.
• The gradient of a scalar field ∇Φ is the direction of steepest ascent from any point in the field.
• The change to the field.

Conservative Fields

• V is the gradient of some scalar field Φ.
• This is called conservative, and the scalar field Φ is the potential of the conservative field.

Example gradient

• For a Function V = x²yz4
• ∇ = 2xyz4i + x²z4j + 4x²yz³k

Divergence of a Vector Field

• The scalar product of nabla and vector field is divergence of vector field V.
• ∇ · V = div V
• The divergence is a measure of the flux, or is it acting like a "source".
• Imagine water flowing through a pipe (vector field).
• If there are no other sources nor drains, divergence is zero.
• Positive divergence: additional pipe adds water to the system.
• Negative divergence: additional pipe drains water from the system.
• Electric point charge: field lines extend to infinity.

Curl of a Vector Field

• The cross product of nabla operator and vector field V.
• ∇ × V = curl V
• Curl describes the "whirliness" of a vector field.
• Flow of water: Vector field is related to vortices left after paddles are used.
• Curl=angular velocity of the water around any point.