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MESF 5010 Quiz 1 Lecturer: Sherry Chen 2024 Fall Throughout this homework, ple...

Oct 4, 2024

Solution by Steps

step 1

For the expression

a_{m} b_{n} \delta_{m p}

, we identify it as a tensor. The Kronecker delta

\delta_{m p}

acts as a selector, thus the full expression is

a_{m} b_{n} \delta_{m p} = a_{p} b_{n}

step 2

For the expression

A_{m n} B_{n m} x_{p}

, we recognize it as a scalar product of two tensors. The full expression is

A_{m n} B_{n m} x_{p} = \sum_{n} A_{m n} B_{n m} x_{p}

step 3

The expression

\delta_{i i} A_{j j}

simplifies to

3 A_{j j}

since

\delta_{i i} = 3

in three dimensions. The full expression is

3 A_{j j}

step 4

The expression

a_{i} b_{j} c_{i} d_{j}

is a product of vectors and can be expressed as a tensor. The full expression is

a_{i} b_{j} c_{i} d_{j}

Answer

The tensors and their full expressions are: 1.

a_{m} b_{n} \delta_{m p} = a_{p} b_{n}

A_{m n} B_{n m} x_{p} = \sum_{n} A_{m n} B_{n m} x_{p}

3 A_{j j}

a_{i} b_{j} c_{i} d_{j}

Key Concept

Understanding tensors and their representations in terms of indices and operations is crucial in tensor calculus.

Explanation

The answers provide the identification of each tensor and its corresponding full expression, which is essential for further calculations in tensor analysis.