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Scalar, Vectors, and Tensors

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Dot product of a tensor with another tensor. Dot product of a vector ... Curvilinear Coordinates. Cylindrical coordinate system. Spherical coordinate system ... – PowerPoint PPT presentation

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Title: Scalar, Vectors, and Tensors


1
Scalar, Vectors, and Tensors
  • Lecture Notes 3

2
Scalar

3
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.
5
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

6
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

7
Curvilinear Coordinates
  • Cylindrical coordinate system
  • Spherical coordinate system

8
Vector and Tensor Integral Theorems
  • Gauss divergence theorem
  • Leibnitz formula
  • Substantial derivative
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