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3. VECTOR ANALYSIS
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Chapter 3: Vector Analysis
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Laws of Vector Algebra
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Properties of Vector Operations
Equality of Two Vectors
Commutative property
Equal Vectors!
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Vector Multiplication: Scalar Product or ”Dot Product”
Hence:
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Vector Multiplication: Vector Product or ”Cross Product”
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Example
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Notes!
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Example:
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Example
Use
Example
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Coordinate Systems
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Cartesian Coordinate System
Differential length vector
Differential area vectors
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Cylindrical Coordinate System
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Independent of φ. q
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Independent of φ. q
We assume A is all around the
circle independent of φ
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Note that the interested area is on the φ-z surface dsr
Notes!
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Note that we use dsr = r dφ dz
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Spherical Coordinate System
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Notes!
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Coordinate Transformations: Coordinates
To solve a problem, we select the coordinate system that best fits its geometry
Sometimes we need to transform between coordinate systems Transforming a point Transforming a vector coordinates
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We have a point and a vector!
Example
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Example
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Another example:
Convert from cylindrical to spherical
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Next….
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Gradient of A Scalar Field
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Gradient ( cont.)
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Gradient ( cont.)
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Divergence of a Vector Field
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Divergence Theorem
Useful tool for converting integration over a volume to one over the surface enclosing that volume, and vice versa
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Curl of a Vector Field
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Stokes’s Theorem
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Laplacian Operator
Laplacian of a Scalar Field
Laplacian of a Vector Field
Useful Relation