MSE 598X MICROHYDRODYNAMICS OF SOFT MATERIALS Problem #1 PROBLEM SET #1 DUE: SEPTEMBER 7, 2023 (a) Given that A is a matrix and x is a vector, what is the expression (A x). (A x) when written in index notation? Is the result a scalar, vector, or tensor? (b) Use index notation to prove the following: ATA = (ATA)T (c) Use index notation to prove the following identity from vector calculus: curl (x)=(div) (div) + grad - grad Problem #2 Find a simplified expression for Eijkfij!. Problem #3 (a) Using index notation, find a simplified expression for: (b) Using index notation, show that: Problem #4 D2 8 2 , 3 (1) (2) Integrate the tensor of unit normals n,n; over the surface of a cylinder with radius R and length L. The cylinder is pointed along the direction u. Here, the vector n, is a unit normal vector to the cylinder surface. You can assume that the cylinder has no end caps. (Hint: The answer will not strictly be isotropic, because the cylinder is pointing in direction us.) Problem #5 Microfluidic applications often involve viscous-dominated flows (low Reynolds number) of Newtonian fluids. Here, the local fluid velocity v(x) can be expressed as a linear function of the position vector such that: v(x) = A.x where A is the (constant) velocity gradient tensor. For "mixed" two-dimensional linear flows, the velocity can be expressed as: v=j -X a 1 (3)