- 4. (15 points total) In fluid mechanics, the Cauchy stress tensor 7 is a 3...

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- 4. (15 points total) In fluid mechanics, the Cauchy stress tensor 7 is a 3 3 matrix that can predict the force of fluid pushing against a flat surface. In particular, if x [x1 x2 x3] represents the normal vector to a flat surface in 3-dimensional space such that ||x|| = == x + x + x is the area of the surface, then the matrix multiplication 7x represents the force vector on that surface. Suppose for a given fluid configuration that the Cauchy stress tensor is 2 1 1 T = 1 2 1 1 2 Use this stress tensor to answer the following questions. (a) (3 points) Compute the force vector on a surface with normal vector x = [1; 1; 1]. (b) (4 points) Suppose the force vector on an unknown surface is f = [0; 0; 1]. Derive a 3 3 system of equations for the components x; of the normal vector to the unknown surface. Express your equations in linear system notation, in matrix-vector form, and as an augmented matrix. (c) (3 points) Reduce your linear system in (b) to triangular form by Gaussian elimination. (You can use the augmented matrix of the system if you prefer.) (d) (3 points) Solve the triangular system from (c) by back substitution. (e) (2 points) What is the normal vector of the unknown surface from (b)? What is its surface area? - 4. (15 points total) In fluid mechanics, the Cauchy stress tensor 7 is a 3 3 matrix that can predict the force of fluid pushing against a flat surface. In particular, if x [x1 x2 x3] represents the normal vector to a flat surface in 3-dimensional space such that ||x|| = == x + x + x is the area of the surface, then the matrix multiplication 7x represents the force vector on that surface. Suppose for a given fluid configuration that the Cauchy stress tensor is 2 1 1 T = 1 2 1 1 2 Use this stress tensor to answer the following questions. (a) (3 points) Compute the force vector on a surface with normal vector x = [1; 1; 1]. (b) (4 points) Suppose the force vector on an unknown surface is f = [0; 0; 1]. Derive a 3 3 system of equations for the components x; of the normal vector to the unknown surface. Express your equations in linear system notation, in matrix-vector form, and as an augmented matrix. (c) (3 points) Reduce your linear system in (b) to triangular form by Gaussian elimination. (You can use the augmented matrix of the system if you prefer.) (d) (3 points) Solve the triangular system from (c) by back substitution. (e) (2 points) What is the normal vector of the unknown surface from (b)? What is its surface area?