Explore BrainMass

Explore BrainMass

    Force as the Gradient of Potential Energy

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    See attached file for full problem description.

    1) 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.

    2) 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).

    3) Calculate the gradient of the following functions, f(x, y, z): (a) f = x2 + z3, (b) f = ky, where k is a constant, (c) f = r sqrt(x2 + y2 + z2), (d) f = 1/r.

    4) Calculate the gradient 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.

    5) Prove that if f(r) and g(r) are any two scalar functions of r, then (fg) = f  g + g f.

    6) 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?

    © BrainMass Inc. brainmass.com June 3, 2020, 7:33 pm ad1c9bdddf
    https://brainmass.com/math/derivatives/force-as-the-gradient-of-potential-energy-103757

    Attachments

    Solution Summary

    Word document find partial derivatives, calculates gradients and proves that two functions are scalar.

    $2.19

    ADVERTISEMENT