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ATP Class 05 Vectors and Tensors 3 (1).pptx
- 1. Vectors and Tensors Class 5: Tensors and Tensor Algebra Reading: BSL, Appendix A
- 2. Logistics Exams and Grading Scheme Exam 2 March 2 Thursday 4 PM Exam 3 April 24 Monday 2:30 PM Exam 1 February 2 Thursday 4 PM Attendance 10% 20% 30% 40% 2
- 3. You’re Here… Motivation and Overview Momentum Transport: Molecular Level Notation Vector Algebr a Tensor Algebra Momentum Transport: Microscopic Level Calculus Exam 1 Feb 2
- 4. Structure • Molecular mechanisms • Vectors and tensors • Shell balances • Equations of change • Steady, unidirectional flows • Dimensional analysis • Time-dependence • 2D flows • Stream Function • Potential flow • Boundary layer theory • Turbulence • Friction factors • Macroscopic balances • Non-Newtonian fluids Section 2 ODEs Section 3 PDEs Section 4 And more… Section 1 Fundamental
- 11. Review- Relationship #2 11 22 12 21
- 13. Review- The Kronecker Delta: Simplifier
- 14. Review- The Kronecker Delta: Simplifier Step 1: Delete the Kronecker delta Step 2: Delete the summation over one of the two indexes of the Kronecker delta Step 3: In the remaining terms, replace the deleted index with the other one
- 15. Review- The Kronecker Delta: Simplifier Step 1: Delete the Kronecker delta Step 2: Delete the summation over one of the two indexes of the Kronecker delta Step 3: In the remaining terms, replace the deleted index with the other one
- 16. Review- The Kronecker Delta: Simplifier Step 1: Delete the Kronecker delta Step 2: Delete the summation over one of the two indexes of the Kronecker delta Step 3: In the remaining terms, replace the deleted index with the other one
- 17. Review- The Kronecker Delta: Simplifier Step 1: Delete the Kronecker delta Step 2: Delete the summation over one of the two indexes of the Kronecker delta Step 3: In the remaining terms, replace the deleted index with the other one
- 18. Review- The Kronecker Delta: Simplifier Step 1: Delete the Kronecker delta Step 2: Delete the summation over one of the two indexes of the Kronecker delta Step 3: In the remaining terms, replace the deleted index with the other one
- 19. Review- The Kronecker Delta: Simplifier Step 1: Delete the Kronecker delta Step 2: Delete the summation over one of the two indexes of the Kronecker delta Step 3: In the remaining terms, replace the deleted index with the other one
- 20. Review- Unit Vectors x y z
- 23. Review- Addition and Subtraction
- 26. Outline • Vector identity • A new vector multiplication • Unit dyads • Tensors: Component form • Tensor operations • Addition and subtraction • Multiplication
- 27. • Vector Identity: Example 1 Vector Vector Scalar Scalar
- 28. Vector Identity: Example 1, LHS
- 29. Vector Identity: Example 1, LHS
- 30. Vector Identity: Example 1, LHS
- 31. Vector Identity: Example 1, LHS
- 32. Vector Identity: Example 1, LHS
- 33. Vector Identity: Example 1, LHS
- 34. Vector Identity: Example 1, LHS
- 35. Vector Identity: Example 1, LHS
- 36. Vector Identity: Example 1, LHS
- 37. Vector Identity: Example 1, LHS
- 38. Vector Identity: Example 1, LHS
- 39. Vector Identity: Example 1, LHS
- 40. Vector Identity: Example 1, LHS
- 41. Vector Identity: Example 1, LHS
- 42. Vector Identity: Example 1, LHS
- 43. Vector Identity: Example 1, LHS
- 44. Vector Identity: Example 1, LHS
- 45. Vector Identity: Example 1, LHS
- 46. Vector Identity: Example 1, LHS
- 47. Vector Identity: Example 1, LHS
- 48. Vector Identity: Example 1, LHS
- 49. Vector Identity: Example 1, LHS
- 50. Vector Identity: Example 1, LHS
- 51. Vector Identity: Example 1, LHS
- 52. Vector Identity: Example 1, LHS
- 53. Vector Identity: Example 1, LHS
- 54. Vector Identity: Example 1, LHS
- 55. Vector Identity: Example 1, LHS
- 56. Vector Identity: Example 1, LHS
- 57. Vector Identity: Example 1, LHS
- 58. Vector Identity: Example 1, LHS
- 59. Vector Identity: Example 1, LHS 11 22 12 21
- 60. Vector Identity: Example 1, LHS 11 22 12 21
- 61. Vector Identity: Example 1, LHS
- 62. Vector Identity: Example 1, LHS
- 79. Vector Identity: Example 1, LHS
- 80. Vector Identity: Example 1, LHS
- 81. Vector Identity: Example 1, LHS
- 82. Vector Identity: Example 1, LHS
- 83. Vector Identity: Example 1, RHS
- 84. Vector Identity: Example 1, RHS
- 85. Vector Identity: Example 1, RHS
- 86. Vector Identity: Example 1, RHS LHS = RHS
- 90. Vector Multiplication: Dyadic Product “ordered pair of coordinate directions” 9
- 91. Vector Multiplication: Unit Dyads
- 92. Vector Multiplication: Unit Dyads
- 93. Unit Dyads: Dot Product with a Unit Vector
- 94. Unit Dyads: Dot Product with a Unit Vector
- 95. Unit Dyads: Dot Product with a Unit Vector
- 96. Unit Dyads: Dot Product with a Unit Vector
- 97. Unit Dyads: Dot Product with a Unit Vector Vector
- 98. Unit Dyads: Dot Product with a Unit Vector
- 99. Unit Dyads: Dot Product with a Unit Vector
- 100. Unit Dyads: Dot Product with a Unit Vector Vector
- 101. Unit Dyads: Dot Product
- 102. Unit Dyads: Dot Product
- 103. Unit Dyads: Dot Product Tenso r
- 104. Unit Dyads: Cross Product with a Unit Vector
- 105. Unit Dyads: Cross Product with a Unit Vector
- 106. Unit Dyads: Cross Product with a Unit Vector
- 107. Unit Dyads: Cross Product with a Unit Vector
- 108. Unit Dyads: Cross Product with a Unit Vector
- 109. Unit Dyads: Cross Product with a Unit Vector Tenso r
- 110. Unit Dyads: Double Dot Product
- 111. Unit Dyads: Scalar Double Product
- 112. Unit Dyads: Scalar Double Product
- 113. Unit Dyads: Scalar Double Product Scalar
- 119. Tensors: Addition and Subtraction
- 120. Tensors: Addition and Subtraction
- 121. Tensors: Addition and Subtraction Tenso r
- 122. Tensors: Multiplications Scalars Vectors Tensors Tensors Dot Cross Dyadic Dot Cross Dyadic Double Dot
- 123. Tensors: Multiplication with a Scalar
- 124. Tensors: Multiplication with a Scalar
- 125. Tensors: Multiplication with a Scalar Tenso r
- 126. Tensors: Dot Product with a Vector
- 127. Tensors: Dot Product with a Vector
- 128. Tensors: Dot Product with a Vector
- 129. Tensors: Dot Product with a Vector
- 130. Tensors: Dot Product with a Vector
- 131. Tensors: Dot Product with a Vector Vector
- 132. Tensors: Dot Product with a Vector
- 133. Tensors: Dot Product with a Vector
- 134. Tensors: Cross Product with a Vector
- 135. Tensors: Cross Product with a Vector
- 136. Tensors: Cross Product with a Vector
- 137. Tensors: Cross Product with a Vector
- 138. Tensors: Cross Product with a Vector
- 139. Tensors: Cross Product with a Vector
- 140. Tensors: Cross Product with a Vector
- 141. Tensors: Single Dot Product
- 142. Tensors: Single Dot Product
- 143. Tensors: Single Dot Product
- 144. Tensors: Single Dot Product
- 145. Tensors: Single Dot Product
- 146. Tensors: Single Dot Product
- 147. Tensors: Single Dot Product
- 148. Tensors: Single Dot Product Tenso r
- 149. Tensors: Double Dot Product
- 150. Tensors: Double Dot Product
- 151. Tensors: Double Dot Product
- 152. Tensors: Double Dot Product
- 153. Tensors: Double Dot Product
- 154. Tensors: Double Dot Product Scalar
- 155. Tensor Identity: Example 1 • Tensor Tensor Scalar Vector Vector