Scalar, Vectors, and Tensors

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Scalar, Vectors, and Tensors

• Lecture Notes 3

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Scalar

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Vectors

• Coordinate system
• Scalar-Vector operation
• Vector-Vector operations
• Dot (scalar) product
• Cross (vector) product
• Vector addition
• Coordinate transformation
• Einstein summation notation for vectors
• Dot product
• Koneke delta
• Cross product
• Epsilon permutation symbol

4
Tensors
A tensor, like a vector, is a mathematical
object with a physical meaning. Unlike scalars
and vectors that are physical entities, tensors
are mathematical entities and are operators.
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Tensors

• Dyad or dyadic product
• Tensor addition
• Dot product of a tensor with another tensor
• Dot product of a vector with a tensor
• Tensor scalar product
• Tensor Einstein notation
• Transpose of a tensor
• Linear vector functions
• Identity tensor
• Zero tensor
• Magnitude of a tensor
• Symmetric tensor
• Anti-symmetric tensor
• Invariants of a tensor
• Trace of a tensor
• Inverse of a tensor

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Vector and Tensor Operations

• Del or nabla operator
• Del operation on a scalar
• Del operation on a vector
• Del operation on a tensor
• Divergence of a vector
• Divergence of a tensor
• Laplacian of a scalar
• Laplacian of a vector
• Laplacian of a tensor

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Curvilinear Coordinates

• Cylindrical coordinate system
• Spherical coordinate system

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Vector and Tensor Integral Theorems

• Gauss divergence theorem
• Leibnitz formula
• Substantial derivative