Vectors and Vector Analysis

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Vectors and Vector Analysis

• Scalar Representation of a quantity using only
  a number (e.g., temperature).
• Vector Representation of a quantity that has
  both magnitude and direction (e.g., wind
  velocity).
• Unit Vectors Vectors with unit length (i.e.,
  magnitude 1) that are parallel to the
  coordinate axes.

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Representation of Vectors
Geometric
Analytic
(2-Dimensional)
(3-Dimensional)
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Magnitude of a Vector
Example
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Basic Vector Operations
Multiplication of a vector by a scalar
(change in magnitude without change in direction)
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Begin with
Vector Addition
Vector Subtraction
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Scalar or Dot Product
Note
If
then
is a scalar quantity
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Vector or Cross Product
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Vector or Cross Product
Right-hand rule
is a vector quantity
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Triple Products (Caution!)
is undefined.
is undefined.
is undefined.
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Uses of the del (or gradient) operator
If f f(x,y,z,t) is a scalar function, then
indicates gradientis computed in 3dimensions
horizontal gradient
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Eulers relation (expansion of total derivative)
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Divergence of a Vector
(Divergence is a scalar quantity.)
Example (divergence of the wind)
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Curl of a Vector
(Curl is a vector quantity.)
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Laplacian of a Scalar
(The Laplacian is a scalar quantity.)