Once we have coordinates, we can establish basis vectors e; and can break vectors and other...

  

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Once we have coordinates, we can establish basis vectors e; and can break vectors and other geometric objects into components, with each component labeled by an index. To work with these objects we therefore need to know how to work with indices and different coordinate systems appropriately. a. Compute 7- 7, where i = w = (1, 1, 1) in a) Cartesian and b) spherical polar coordinates. Both vectors are in T, (the tangent space) at some arbitrary point p. b. Why do we even care about the summation convention where summed over indices have to be one up and one down? Let's do an example that will illuminate this point. Consider the vector = (1,0) at the point (1, 7/2) in the Euclidean plane in polar coordinates, (a) Draw this vector. (b) Construct va (c) Apply the summation convention to calculate the scalar vav". (d) Is vau independent of coordinates? Prove your answer. (e) Now let's apply the "incorrect" summation convention: calculate v"" by summing over the index a. This is also just a number. Is this quantity independent of coordinates? Prove your answer. (f) Why then do we require for physics that repeated indices that are summed over must be one up/one down? c. Which of the following expressions are incorrect according to the summation convention? (a) AngB Ags (b) gosga (c) Rasgos (d) Tg d. Let aan and bay be two symmetric rank 2 tensors. Remember, a symmetric rank 2 tensor is one where aas = aga. Furthermore, let the two tensors satisfy the relation: dopby5 – aasbar + a3,bas – aysbas = 0. Prove that ans = fbas where f is a scalar. Once we have coordinates, we can establish basis vectors e; and can break vectors and other geometric objects into components, with each component labeled by an index. To work with these objects we therefore need to know how to work with indices and different coordinate systems appropriately. a. Compute 7- 7, where i = w = (1, 1, 1) in a) Cartesian and b) spherical polar coordinates. Both vectors are in T, (the tangent space) at some arbitrary point p. b. Why do we even care about the summation convention where summed over indices have to be one up and one down? Let's do an example that will illuminate this point. Consider the vector = (1,0) at the point (1, 7/2) in the Euclidean plane in polar coordinates, (a) Draw this vector. (b) Construct va (c) Apply the summation convention to calculate the scalar vav". (d) Is vau independent of coordinates? Prove your answer. (e) Now let's apply the "incorrect" summation convention: calculate v"" by summing over the index a. This is also just a number. Is this quantity independent of coordinates? Prove your answer. (f) Why then do we require for physics that repeated indices that are summed over must be one up/one down? c. Which of the following expressions are incorrect according to the summation convention? (a) AngB Ags (b) gosga (c) Rasgos (d) Tg d. Let aan and bay be two symmetric rank 2 tensors. Remember, a symmetric rank 2 tensor is one where aas = aga. Furthermore, let the two tensors satisfy the relation: dopby5 – aasbar + a3,bas – aysbas = 0. Prove that ans = fbas where f is a scalar.

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