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Knowledge
The chapter introduces the fundamental concepts of vectors and tensors, covering their definitions, representations, and mathematical operations including the dot product, cross product, and tensor product. It also discusses second-order tensors, their operations with vectors, and how to extract coefficients from tensor representations. Finally, the chapter explains rotation tensors and their properties, emphasizing the importance of these concepts in solid mechanics.
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Sections
References
ch1.pdfClass Notes
Memorization
What we have learnt
- Vectors have both magnitude...
- The dot product of vectors ...
- Tensors can be of various o...
Final Test
Revision Tests
What we have learnt
- Vectors have both magnitude and direction and can be represented in various coordinate systems.
- The dot product of vectors yields a scalar, while the cross product results in a vector.
- Tensors can be of various orders, with second-order tensors resulting from the tensor product of two vectors.
Key Concepts
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Term: Vector
Definition: A quantity that has both magnitude and direction, represented as an arrow in space.
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Term: Dot Product
Definition: An operation that takes two vectors and returns a scalar, calculated as the summation of the product of their corresponding components.
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Term: Cross Product
Definition: An operation that takes two vectors and returns a vector that is perpendicular to the plane formed by the original vectors.
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Term: Tensor
Definition: An algebraic object that can be represented as a multi-dimensional array and has properties that are independent of the coordinate system.
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Term: Rotation Tensor
Definition: A special tensor that applies a rotation to vectors and other tensors while preserving their magnitudes, represented in an orthonormal matrix form.