1. Vector Analysis 1.1 Vector Algebra. 1.1.1 Vector operations A scalar has a magnitude (mass, time, temperature, charge). A vector has a magnitude (its.

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Presentation transcript:

1. Vector Analysis 1.1 Vector Algebra

1.1.1 Vector operations A scalar has a magnitude (mass, time, temperature, charge). A vector has a magnitude (its length) and a direction. Examples: velocity, force, momentum, field strength. Boldface letters denote vectors. On the blackboard I use. Unit vectors are denoted by

Vectors have no location. -A Vector field A(r)

addition of two vectors: A+B multiplication by a scalar: aA

dot product (scalar, inner): if parallel if perpendicular Example work

Example 1.1

cross product (vector, outer): is the unit vector perpendicular to the AB-plane. form a right-handed system. is the area of the parallelogram. Example: angular momentum

1.1.2 Component Form 1: x, 2: y, 3: z components: basis:

common notation: Kronecker symbol Properties of the basis

Levi-Civita symbol

Example 1.2

1.1.3 Triple Products scalar triple product: volume

vector triple product: bac - cab rule Higher order products by repeated bac-cab and symmetries of the scalar triple product.

1.1.4 Notation

1.1.5 How Vectors Transform Rotation about the x-axis: In general