Tensors for Beginners 1: Forward and Backward Transformations (REMAKE)

Summary

In this video, the speaker introduces forward and backward transformations, key concepts in tensor mathematics. The forward transformation moves between old and new coordinate systems by expressing new basis vectors as linear combinations of old ones, represented in a matrix form. The backward transformation does the reverse, and the matrices F and B are inverses of each other, illustrated through matrix multiplication. The video also touches on generalizing the process to n dimensions and introduces the concept of the Kronecker Delta. The next video will explore vector components and how these transformations apply to them.

Takeaways

• 😀 Tensors are invariant under a coordinate system change, so understanding transformations is key to working with them.
• 😀 The forward transformation allows us to convert vectors from an old basis to a new basis by expressing new basis vectors as linear combinations of the old ones.
• 😀 In two dimensions, the forward transformation involves building the new basis vectors (E1 tilde, E2 tilde) from the old basis vectors (E1, E2).
• 😀 A matrix, F, is used to represent the forward transformation, where the basis vectors are written as rows, and matrix multiplication helps find the transformed vectors.
• 😀 The backward transformation is essentially the reverse process, building the old basis vectors using the new ones. The matrix for this transformation is called B.
• 😀 The matrices F (forward) and B (backward) are inverses of each other, meaning B = F inverse.
• 😀 When multiplying the forward and backward matrices (F and B), the result is the identity matrix, confirming they are indeed inverses.
• 😀 In N-dimensional space, the same principles apply: we use summation to express new basis vectors as linear combinations of old ones, represented by an N x N matrix, F.
• 😀 The backward transformation in N dimensions is similarly represented by matrix B, and both transformations can be generalized to higher dimensions.
• 😀 The Kronecker Delta is used to represent the behavior of the identity matrix, with ones on the diagonal and zeros elsewhere, a concept that often appears in tensor transformations.
• 😀 The forward and backward transformations help us build vectors in different coordinate systems, and understanding these processes is essential for working with tensors in higher-dimensional spaces.

Q & A

What is the primary purpose of the forward transformation in tensor analysis?

The forward transformation is used to move from an old coordinate system (with old basis vectors) to a new coordinate system (with new basis vectors). It achieves this by expressing the new basis vectors as linear combinations of the old ones.

How do you represent the forward transformation using matrices?

The forward transformation can be represented by a matrix (denoted F), where the coefficients for the linear combinations of the old basis vectors to construct the new ones are placed in a 2x2 (or NxN for higher dimensions) matrix.

What is the backward transformation and how is it different from the forward transformation?

The backward transformation is the reverse of the forward transformation. It constructs the old basis vectors using the new basis vectors. The matrix representing the backward transformation is the inverse of the forward transformation matrix.

What does the multiplication of the forward and backward matrices yield, and why is this significant?

Multiplying the forward matrix (F) and the backward matrix (B) yields the identity matrix. This is significant because it confirms that F and B are inverses of each other, meaning that applying the forward transformation and then the backward transformation (or vice versa) will bring you back to the original coordinate system.

How do the forward and backward transformations generalize to higher dimensions?

In higher dimensions (N dimensions), the forward and backward transformations work the same way but involve summations over multiple basis vectors. The transformation matrices become NxN matrices, and the same principles of linear combinations and matrix multiplication apply.

What role does the Chronicler Delta play in tensor transformations?

The Chronicler Delta symbol represents the identity matrix behavior, where it equals 1 when the row index equals the column index (i.e., on the main diagonal) and 0 elsewhere. It is used to simplify expressions and equations in tensor analysis, particularly when demonstrating matrix multiplication results like the identity matrix.

Why is the arrangement of numbers in the forward transformation matrix flipped about the diagonal compared to the equations?

The numbers in the forward transformation matrix are flipped about the diagonal because, in this video series, basis vectors are written as rows, and matrix multiplication is done from the left. This arrangement ensures that multiplying the row vector by the matrix correctly yields the new basis vectors.

What is the significance of the identity matrix in the context of forward and backward transformations?

The identity matrix represents the idea that applying both the forward and backward transformations sequentially results in no change to the coordinate system. It verifies that the forward and backward transformations are indeed inverses of each other.

How are the forward and backward transformations related to the components of a vector in different coordinate systems?

The forward and backward transformations allow you to convert vector components from one coordinate system to another. The components of a vector are usually written as columns, and these components can be transformed using the forward or backward matrices to switch between coordinate systems.

What is the purpose of using the notation with summation indices in the general form of the transformations?

The summation notation allows for a compact and generalized way to express the linear combinations of basis vectors in the forward and backward transformations. It helps in representing these transformations in N-dimensional spaces without having to write out each equation individually.