In terms of spherical coordinates (p. theta, psi) on R^3, the Laplacian on the u

Question

In terms of spherical coordinates (p. theta, psi) on R^3, the Laplacian on the unit sphere S^2 of a function f(theta, psi) is given by delta f (theta, psi) = 1/sin psi [(sin psi f_psi)_psi + f_theta theta/sin psi]. (a) Show that the restriction of a harmonic homogeneous polynomial of degree 1 on R^3, satisfies the equation delta f + 2f = 0 on the unit sphere S^2. (b) Choose any harmonic homogeneous polynomial on R^3 of degree 2 (except f 0) and show that your choice satisfies delta f + 6 f = 0 on S^2.

Explanation / Answer

The Laplace operator or laplacian is a differential operator given by the divergence of the gradient of a function of Euclidean space. It is usually denoted by the symbol reverse delta.

A homogeneous polynomial is a polynomial whose non zero terms all have the same degree. For example is a homogeneous polynomial of degree 5, in two variables, the sum of the exponents in each term is always 5.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.