Solution Found!
Let n1 and n2 be any two unit vectors and P a point in a
Chapter 16, Problem 16.19(choose chapter or problem)
Let n1 and n2 be any two unit vectors and P a point in a continuous medium. F(n1 dA) is the surface force on a small area dA at P with unit outward normal n1, so n2 F(ni dA) is the component of that force in the direction of n2. Prove Cauchy's reciprocal theorem that n2 F(n1 dA) = n1 F(n2 dA).
Questions & Answers
QUESTION:
Let n1 and n2 be any two unit vectors and P a point in a continuous medium. F(n1 dA) is the surface force on a small area dA at P with unit outward normal n1, so n2 F(ni dA) is the component of that force in the direction of n2. Prove Cauchy's reciprocal theorem that n2 F(n1 dA) = n1 F(n2 dA).
ANSWER:Step 1 of 3
The surface force on a small surface element \(\mathbf{n}_1dA\) is \(\mathbf{F}(\mathbf{n}_1dA) = \mathbf{\Sigma n}_1dA\) and the component of this force in the direction of \(\mathbf{n}_2\) is \(\mathbf{n}_2\cdot \left ( \mathbf{\Sigma n}_1 \right )dA\).