Vector Analysis Introduction

Scalar Field (Potential Function)

• Definition: Assigns a scalar value (real number) to each point in space.
• Example: Temperature field in space.
• Level Surface: Surface where the scalar field is constant (e.g., 20 degrees).

Vector Field

• Definition: Assigns a vector to each point in space, includes direction.
• Example: Force field (force has direction and magnitude).

Gradient

• Definition: Perpendicular to the level surface and points in the direction of the greatest increase of the scalar field.
• Calculation: With nabla operator and partial derivatives of the potential function.

Directional Derivative

• Definition: Measure of the change in the function value of phi when moving in the direction of a unit vector.
• Calculation: Product of gradient and unit vector in a specific direction.

Divergence of a Vector Field

• Definition: Describes the source strength or source density (scalar value).
• Calculation: Nabla operator multiplied with vector field, derivatives added.
• Values:

  • 0: source

  • =0: source-free
  • <0: sink

Curl of a Vector Field

• Definition: Describes the vorticity density.
• Special Case: If curl = 0, the vector field is irrotational and conservative.
• Calculation: Cross product of nabla operator and vector field.
• 2D Vector Field: Only the z-component is relevant (y derivative with respect to x - x derivative with respect to y).

Relationships

• Conservative Vector Field: Curl = 0, thus a potential function exists.
• Calculation: Gradient of the potential function results in the original vector field.

Outlook

• Further calculation methods will be explained in the following videos.
• Click through playlists for more details.