Question

1. The general form for homogeneous polynomial of degree 3 is P3 = ax^3 + bx^2y + cxy^2 + dy^3 where a, b, c, d are constants. (i) [4pts] Find one explicit (particular) P3 that is a harmonic function, i.e., a function statisfies ?P3 = 0. Note: find an explicit one (like the one P2 = x^2 - y^2 in lecture), not just derive some relation for coefficients. (ii) [1pts] What is the dimension of the space of all P3 that are harmonic functions.

          1. The general form for homogeneous polynomial of degree 3 is
P3 = ax^3 + bx^2y + cxy^2 + dy^3
where a, b, c, d are constants.
(i) [4pts] Find one explicit (particular) P3 that is a harmonic function, i.e.,
a function statisfies
?P3 = 0.
Note: find an explicit one (like the one P2 = x^2 - y^2 in lecture), not just
derive some relation for coefficients.
(ii) [1pts] What is the dimension of the space of all P3 that are harmonic
functions.

1. The general form for homogeneous polynomial of degree 3 is
P3 = ax^3 + bx^2y + cxy^2 + dy^3
where a, b, c, d are constants.
(i) [4pts] Find one explicit (particular) P3 that is a harmonic function, i.e.,
a function statisfies
?P3 = 0.
Note: find an explicit one (like the one P2 = x^2 - y^2 in lecture), not just
derive some relation for coefficients.
(ii) [1pts] What is the dimension of the space of all P3 that are harmonic
functions.

Added by Janice V.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Jerelyn Nevil

Step 1

A harmonic function satisfies Laplace's equation, which means that the sum of the second partial derivatives with respect to each variable is zero: $$\nabla^2 P_3 = \frac{\partial^2 P_3}{\partial x^2} + \frac{\partial^2 P_3}{\partial y^2} = 0$$ Let's consider Show more…

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The general form for homogeneous polynomial of degree 3 is P3 = ax^3 + bx^2y + cxy^2 + dy^3 where a, b, c, d are constants. (i) Find one explicit (particular) P3 that is a harmonic function, i.e., a function satisfies ΔP3 = 0. Note: find an explicit one (like the one P2 = x^2 - y^2 in lecture), not just derive some relation for coefficients. (ii) What is the dimension of the space of all P3 that are harmonic functions.