Assignment:
Q1) Find the partial derivatives with respect to x, y, and z of the following functions: (a) f(x, y, z) = ax2 + bxy + cy2, (b) g(x, y, z) = sin(axyz2), (c) h(x, y, z) = aexy/z^2, where a, b, and c are constants.
Q2) Find the partial derivatives with respect to x, y, and z of the following functions: (a) f(x, y, z) = ay2 + 2byz + cz2, (b) g(x, y, z) = cos(axy2z3), (c) h(x, y, z) = ar, where a, b, and c are constants and r = sqrt(x2 + y2 + z2).
Q3) Calculate the gradient ⊗f of the following functions, f(x, y, z): (a) f = x2 + z3, (b) f = ky, where k is a constant, (c) f = r h sqrt(x2 + y2 + z2), (d) f = 1/r.
Q4) Calculate the gradient ⊗f of the following functions, f(x, y, z): (a) f = ln(r), (b) f = r2, (c) f = g(r), where r = sqrt(x2 + y2 + z2) and g(r) is some unspecified function of r.
Q5) Prove that if f(r) and g(r) are any two scalar functions of r, then ⊗(fg) = f ⊗ g + g ⊗ f.
Q6) If a particle's potential energy is U(r) = k(x2 + y2 + z2), where k is a constant, what is the force on the particle?
Provide complete and step by step solution for the question and show calculations and use formulas.