This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
here is the answer to question 2a: 2a. Show that any function given by = (ax)(by)e^-sqrt(a^2+b^2)z satisfies Laplace's equation in three dimensions: (^2 )/( x^2) + (^2 )/( y^2) + (^2 )/( z^2) = 0. Step 1: Calculate the first and second partial derivatives of with respect to x. Let C_y = (by) and E_z = e^-sqrt(a^2+b^2)z. Then = (ax) C_y E_z. ( )/( x) = a(ax)C_y E_z (^2 )/( x^2) = -a^2(ax)C_y E_z = -a^2 Step 2: Calculate the first and second partial derivatives of with respect to y. Let S_x = (ax) and E_z = e^-sqrt(a^2+b^2)z. Then = S_x (by) E_z. ( )/( y) = S_x (-b(by)) E_z = -b(ax)(by)E_z (^2 )/( y^2) = S_x (-b^2(by)) E_z = -b^2(ax)(by)E_z = -b^2 Step 3: Calculate the first and second partial derivatives of with respect to z. Let S_x = (ax) and C_y = (by). Let k = sqrt(a^2+b^2). Then = S_x C_y e^-kz. ( )/( z) = S_x C_y (-k e^-kz) = -k(ax)(by)e^-kz (^2 )/( z^2) = S_x C_y (-k)(-k e^-kz) = k^2(ax)(by)e^-kz = k^2 Substitute k = sqrt(a^2+b^2): (^2 )/( z^2) = (sqrt(a^2+b^2))^2 = (a^2+b^2) Step 4: Sum the second partial derivatives to check Laplace's equation. (^2 )/( x^2) + (^2 )/( y^2) + (^2 )/( z^2) = -a^2 - b^2 + (a^2+b^2) = (-a^2 - b^2 + a^2 + b^2) = 0 = 0 Since the sum is 0, the function satisfies Laplace's equation. Drop the next question.