Unit 1: Vectors and Ordinary Differential Equations

1. Vectors

Dot (Scalar) Product

The dot product of two vectors A and B is a scalar quantity. It is defined as:

Definition: A ⋅ B = |A| |B| cos(θ)

Where |A| and |B| are the magnitudes of the vectors and θ is the angle between them.

In Cartesian coordinates (A = A_xi + A_yj + A_zk, B = B_xi + B_yj + B_zk):

Formula: A ⋅ B = A_xB_x + A_yB_y + A_zB_z

Properties:

• Commutative: A ⋅ B = B ⋅ A
• Distributive: A ⋅ (B + C) = A ⋅ B + A ⋅ C
• If A ⊥ B (orthogonal), then θ = 90° and A ⋅ B = 0.
• If A || B (parallel), then θ = 0° and A ⋅ B = |A| |B|.
• Self-product: A ⋅ A = |A|² = A_x² + A_y² + A_z²

Physical Example: Work done (W) by a constant force F over a displacement d is W = F ⋅ d.

Cross (Vector) Product

The cross product of two vectors A and B is a new vector that is perpendicular to both A and B.

Its magnitude is:

Magnitude: |A × B| = |A| |B| sin(θ)

The direction is given by the Right-Hand Rule. In Cartesian coordinates, it is calculated using a determinant:

Formula: A × B = | i    j    k |
| A_x   A_y   A_z |
| B_x   B_y   B_z |
= (A_yB_z - A_zB_y)i - (A_xB_z - A_zB_x)j + (A_xB_y - A_yB_x)k

Properties:

• Anti-commutative: A × B = - (B × A)
• Distributive: A × (B + C) = A × B + A × C
• If A || B (parallel), then θ = 0° and A × B = 0.
• Self-product: A × A = 0.

Physical Example: Torque (τ) due to a force F applied at a position r from the pivot is τ = r × F.

Scalar and Vector Triple Products

1. Scalar Triple Product

This product results in a scalar. It is defined as:

Definition: A ⋅ (B × C)

Physical Significance: The absolute value |A ⋅ (B × C)| represents the volume of the parallelepiped formed by the vectors A, B, and C.

It can be calculated using a determinant:

Formula: A ⋅ (B × C) = | A_x   A_y   A_z |
| B_x   B_y   B_z |
| C_x   C_y   C_z |

Property: The dot and cross can be interchanged, and the vectors can be cyclically rotated without changing the value:

A ⋅ (B × C) = B ⋅ (C × A) = C ⋅ (A × B) = (A × B) ⋅ C

2. Vector Triple Product

This product results in a vector. It is defined as:

Definition: A × (B × C)

Physical Significance: The resulting vector lies in the plane formed by vectors B and C.

This is computed using the "BAC-CAB" rule:

Formula: A × (B × C) = B(A ⋅ C) - C(A ⋅ B)

2. Introduction to Vector Calculus

We introduce the vector differential operator Del (∇):

Del Operator: ∇ = i(∂/∂x) + j(∂/∂y) + k(∂/∂z)

This operator can act on scalar fields (like temperature, T(x,y,z)) and vector fields (like electric field, E(x,y,z)).

Gradient (grad)

The gradient acts on a scalar field (f) and produces a vector field.

Definition: grad(f) = ∇f = i(∂f/∂x) + j(∂f/∂y) + k(∂f/∂z)

Physical Significance:

• The vector ∇f points in the direction of the steepest increase of the scalar f.
• The magnitude |∇f| is the rate of that maximum increase.
• Example: If T(x,y,z) is a temperature field, ∇T is a vector pointing towards the hottest direction. The electrostatic field is the negative gradient of the electric potential: E = -∇V.

Divergence (div)

The divergence acts on a vector field (V) and produces a scalar field. It is the dot product of ∇ and V.

Definition: div(V) = ∇ ⋅ V = (∂V_x/∂x) + (∂V_y/∂y) + (∂V_z/∂z)

Physical Significance:

• Divergence measures how much a vector field is "spreading out" or "diverging" from a point.
• If ∇ ⋅ V > 0, the point is a source.
• If ∇ ⋅ V < 0, the point is a sink.
• If ∇ ⋅ V = 0, the field is solenoidal (no sources or sinks, like the magnetic field B, ∇ ⋅ B = 0).
• Example: In fluid flow, ∇ ⋅ v (where v is velocity) measures the rate at which fluid is expanding from a point.

Curl (curl)

The curl acts on a vector field (V) and produces another vector field. It is the cross product of ∇ and V.

Definition: curl(V) = ∇ × V = | i          j          k |
| ∂/∂x     ∂/∂y     ∂/∂z |
| V_x       V_y       V_z |

Physical Significance:

• Curl measures the "circulation" or "rotation" of a vector field at a point.
• If ∇ × V = 0, the field is irrotational or conservative (like an electrostatic field, ∇ × E = 0).
• Example: In fluid flow, ∇ × v represents the vorticity (how a paddlewheel would spin at that point). A magnetic field curls around a current: ∇ × B = μ₀J.

3. Vector Integration

Line, Surface, and Volume Integrals

• Line Integral: Integrates a vector field V along a path C.

  Formula: ∫_C V ⋅ dl

  Example: Work done by a variable force F along a path C is W = ∫_C F ⋅ dl.

• Surface Integral: Integrates a vector field V over a surface S.

  Formula: ∫∫_S V ⋅ dS

  Example: Electric flux (Φ_E) through a surface S is Φ_E = ∫∫_S E ⋅ dS.

• Volume Integral: Integrates a scalar field f over a volume τ.

  Formula: ∫∫∫_τ f dτ

  Example: Total charge (Q) in a volume τ with charge density ρ is Q = ∫∫∫_τ ρ dτ.

Gauss-Divergence Theorem (Statement Only)

This theorem relates a volume integral of the divergence of a vector field to a closed surface integral of the flux of that field.

Theorem Statement: The flux of a vector field V through a closed surface S is equal to the volume integral of the divergence of V over the volume τ enclosed by that surface.

Formula: ∮∮_S V ⋅ dS = ∫∫∫_τ (∇ ⋅ V) dτ

Physical Meaning: The total "stuff" flowing out of a closed surface (left side) is equal to the sum of all the "sources" inside that volume (right side).

Stoke's Theorem (Statement Only)

This theorem relates an open surface integral of the curl of a vector field to a closed line integral of that field around the boundary of the surface.

Theorem Statement: The line integral (circulation) of a vector field V around a closed loop C is equal to the surface integral of the curl of V over any open surface S bounded by that loop.

Formula: ∮_C V ⋅ dl = ∫∫_S (∇ × V) ⋅ dS

Physical Meaning: The total "rotation" of a field over a surface (right side) is equal to the "circulation" of the field around the edge of that surface (left side).

4. Ordinary Differential Equations (ODE)

1st Order Homogeneous Differential Equations

A homogeneous ODE of the first order can be written in the form:

dy/dx = f(y/x)

How to Solve (Separation of Variables):

1. Make the substitution v = y/x, which means y = vx.
2. Differentiate y = vx with respect to x: dy/dx = v + x(dv/dx).
3. Substitute v and dy/dx into the original equation: v + x(dv/dx) = f(v).
4. This new equation is separable: x(dv/dx) = f(v) - v.
5. Separate the variables: dv / (f(v) - v) = dx / x.
6. Integrate both sides: ∫ dv / (f(v) - v) = ∫ dx / x = ln|x| + C.
7. Solve the integral for v, and then substitute back v = y/x to get the final solution for y.

2nd Order Homogeneous Differential Equations with Constant Coefficients

This is a very common type of equation in physics (e.g., in Simple Harmonic Motion). The general form is:

General Form: a(d²y/dx²) + b(dy/dx) + cy = 0

Where a, b, and c are constants.

How to Solve:

1. Assume a solution of the form y = e^mx.
2. Substitute this into the ODE:

   y' = me^mx

   y'' = m²e^mx

   a(m²e^mx) + b(me^mx) + c(e^mx) = 0

3. Factor out e^mx (which is never zero): e^mx (am² + bm + c) = 0
4. This leaves the Auxiliary Equation (or Characteristic Equation):

   Auxiliary Equation: am² + bm + c = 0

5. Solve this quadratic equation for 'm'. This gives two roots, m₁ and m₂. The general solution for y depends on the nature of these roots.

Case 1: Two Real, Distinct Roots (b² - 4ac > 0)

The roots are m₁ and m₂. The general solution is:

Solution: y(x) = C₁e^m₁x + C₂e^m₂x

Case 2: Two Real, Repeated Roots (b² - 4ac = 0)

The roots are m₁ = m₂ = m. The general solution is:

Solution: y(x) = (C₁ + C₂x)e^mx

Case 3: Two Complex Roots (b² - 4ac < 0)

The roots are a complex conjugate pair: m = α ± iβ. The solution is:

y(x) = C₁e^{(α + iβ)x} + C₂e^{(α - iβ)x}

Using Euler's formula (e^iθ = cosθ + isinθ), this simplifies to a more useful form:

Solution: y(x) = e^αx (A cos(βx) + B sin(βx))

Example (SHM): d²x/dt² + ω²x = 0.
Auxiliary equation: m² + ω² = 0.
Roots: m = ±iω. (This is Case 3 with α=0, β=ω).
Solution: x(t) = A cos(ωt) + B sin(ωt).