Unit 5: Vector Calculus I

1. Vector Functions: Limit, Continuity, and Differentiability

A vector-valued function is a rule that assigns a vector to each element in a domain (usually a set of real numbers representing time or a parameter).

1.1 Definition and Component Form

If , , and are real-valued functions of the parameter , then a vector function in 3D space is defined as:

1.2 Limits

The limit of a vector function is defined component-wise. If , then:

Condition: The limit exists only if the limits of all constituent component functions exist.

1.3 Continuity

A vector function is continuous at if:

1. is defined.
2. exists.
3. .

1.4 Differentiability

The derivative of a vector function , denoted by , is defined as:

In component form:

Geometric Interpretation: The vector represents the tangent vector to the curve defined by at point , pointing in the direction of increasing parameter .

2. Length of a Space Curve (Arc Length)

If a curve is defined by for , and the derivatives are continuous, the length of the curve is the integral of the magnitude of the tangent vector.

Formula

Arc Length Function

The arc length function , which measures distance from a fixed starting point , is:

From this, we derive the relationship , which represents speed.

3. Motion of a Body or Particle on a Curve

In physics, if represents the position of a particle at time , we define the kinematic quantities as follows:

3.1 Kinematic Vectors

1. Position Vector:
2. Velocity Vector: (Always tangent to the path)
3. Speed: (Scalar quantity)
4. Acceleration Vector:

3.2 Components of Acceleration

Acceleration does not always point in the direction of motion. It can be resolved into two orthogonal components:

• Tangential Component (): Responsible for changing the speed of the particle.
  
• Normal Component (): Responsible for changing the direction of the particle.
  
  (Where is the curvature)

4. Scalar and Vector Fields

Before discussing derivatives in fields, we distinguish between two types of functions in region :

• Scalar Field: Assigns a real number (scalar) to every point in space. Examples: Temperature , Pressure .
• Vector Field: Assigns a vector to every point in space. Example: Velocity of fluid flow , Gravitational field.

The Del Operator ()

The vector differential operator "del" or "nabla" is defined as:

5. Gradient and Directional Derivative

5.1 Gradient of a Scalar Field

If is a scalar field, the gradient of , denoted or grad , is a vector field:

Physical Significance of Gradient:

1. points in the direction of the maximum rate of increase of the function .
2. The magnitude is the maximum rate of change.
3. is normal (perpendicular) to the level surface .

5.2 Directional Derivative

The rate of change of at a point in the direction of a specific unit vector is the directional derivative :

If given a direction vector that is not a unit vector, normalize it first: .

6. Divergence and Curl of a Vector Field

Let be a vector field.

6.1 Divergence ()

Divergence is the dot product of the Del operator and the vector field. It results in a scalar.

Physical Interpretation:

• Measures the net expansion or compression of a fluid at a point.
• Source: If , fluid is flowing out (expanding).
• Sink: If , fluid is flowing in (compressing).
• Solenoidal: If , the fluid is incompressible (no source or sink).

6.2 Curl ()

Curl is the cross product of the Del operator and the vector field. It results in a vector.

Expansion:

Physical Interpretation:

• Measures the rotation or angular velocity of the field at a point.
• The direction of the Curl vector represents the axis of rotation.
• Irrotational: If , the field is conservative (no rotation).

6.3 Important Vector Identities

1. (Curl of a Gradient is zero).
2. (Divergence of a Curl is zero).
3. (Laplacian of a scalar).