Unit 3: Vector Calculus
Table of Contents
Vector Calculus is built around the vector differential operator, del (∇).
In Cartesian coordinates: ∇ = î (∂/∂x) + ĵ (∂/∂y) + k̂ (∂/∂z)
This operator is not a vector itself, but it acts on scalar or vector fields to produce new fields.
Vector Differentiation
This refers to differentiating a vector function of a single scalar variable, like time (t).
If r(t) = x(t)î + y(t)ĵ + z(t)k̂ is the position vector of a particle:
- Velocity (v): v(t) = dr/dt = (dx/dt)î + (dy/dt)ĵ + (dz/dt)k̂
- Acceleration (a): a(t) = dv/dt = d²r/dt² = (d²x/dt²)î + (d²y/dt²)ĵ + (d²z/dt²)k̂
Differentiation Rules: (f is a scalar, A and B are vectors)
- d/dt (A + B) = dA/dt + dB/dt
- d/dt (f A) = (df/dt)A + f(dA/dt) (Product rule)
- d/dt (A · B) = (dA/dt) · B + A · (dB/dt) (Product rule for dot product)
- d/dt (A × B) = (dA/dt) × B + A × (dB/dt) (Product rule for cross product - order matters!)
Directional Derivatives and Normal Derivatives
Directional Derivative
The directional derivative measures the rate of change of a scalar field f(x,y,z) at a specific point P, in a specific direction.
Let the direction be given by a unit vector û.
The directional derivative of f in the direction û is:
Dûf = (∇f) · û
It is the dot product of the gradient of f (see below) and the unit vector of the direction.
- Step 1: Find the gradient, ∇f.
- Step 2: Find the unit vector û. (If given a vector V, û = V / |V|).
- Step 3: Calculate the dot product ∇f · û.
- Step 4: Evaluate the result at the given point P.
Normal Derivative
The "normal derivative" is simply the directional derivative in the direction normal (perpendicular) to a surface. As we will see, the gradient (∇f) is *already* normal to the level surfaces of f. Therefore, the normal derivative is often used to mean the maximum rate of change, which is just the magnitude of the gradient, |∇f|.
Gradient of a Scalar Field (∇f)
The gradient acts on a scalar field (f) and produces a vector field.
grad(f) = ∇f = (∂f/∂x)î + (∂f/∂y)ĵ + (∂f/∂z)k̂
Geometrical Interpretation of Gradient
The gradient ∇f at a point P has a very important physical meaning:
- Direction: ∇f points in the direction of the steepest ascent (maximum rate of change) of the scalar field f.
- Magnitude: |∇f| is the value of this maximum rate of change.
- Normality: ∇f is always normal (perpendicular) to the level surface (or contour line) f(x,y,z) = constant that passes through the point P.
- Force from Potential: A conservative force F is the negative gradient of its scalar potential energy U.
F = -∇U
- Electric Field from Potential: The electric field E is the negative gradient of the electric potential V.
E = -∇V
Divergence of a Vector Field (∇ · V)
The divergence acts on a vector field (V) and produces a scalar field. It is the dot product of ∇ and V.
If V = Vxî + Vyĵ + Vzk̂:
div(V) = ∇ · V = (∂Vx/∂x) + (∂Vy/∂y) + (∂Vz/∂z)
Physical Interpretation of Divergence
Divergence measures the "outflow" or "source strength" of a vector field from an infinitesimal volume around a point.
- ∇ · V > 0: The point is a source. More field lines "diverge" from this point than enter it. (e.g., a positive charge for an E-field).
- ∇ · V < 0: The point is a sink. More field lines enter this point than leave. (e.g., a negative charge).
- ∇ · V = 0: The field is solenoidal (or "divergence-free"). There are no sources or sinks. The same amount of "flow" enters any volume as leaves it.
- Gauss's Law: ∇ · E = ρ / ε₀ (Charge density ρ is the source of E).
- Gauss's Law for Magnetism: ∇ · B = 0 (There are no magnetic monopoles, so B-fields are always solenoidal).
Curl of a Vector Field (∇ × V)
The curl acts on a vector field (V) and produces another vector field. It is the cross product of ∇ and V.
It is calculated using a determinant:
| î ĵ k̂ | curl(V) = ∇ × V = | ∂/∂x ∂/∂y ∂/∂z | | Vx Vy Vz | = (∂Vz/∂y - ∂Vy/∂z)î - (∂Vz/∂x - ∂Vx/∂z)ĵ + (∂Vy/∂x - ∂Vx/∂y)k̂