Calculating the Laplacian of a Function

The Laplacian of a function is a mathematical operation that involves the second partial derivatives of a function. It is used to determine the rate of change of the gradient of a function at any given point in space. The Laplacian of a function f(x,y,z) is denoted by ∇²f or Δf.

Given Function: f(x,y,z) = x² + 2xy + 3z + 4

Step 1: Find the first partial derivatives of the function with respect to x, y, and z.

fₓ = 2x + 2y
fᵧ = 2x
f_z = 3

Step 2: Find the second partial derivatives of the function with respect to x, y, and z.

fₓₓ = 2
fₓᵧ = 2
fᵧᵧ = 0
f_zz = 0
fₓz = 0
fᵧz = 0

Step 3: Calculate the Laplacian of the function by adding the second partial derivatives.

∇²f = fₓₓ + fₓᵧ + fᵧᵧ + f_zz + fₓz + fᵧz
∇²f = 2 + 2 + 0 + 0 + 0 + 0
∇²f = 4

Therefore, the Laplacian of the function f(x,y,z) = x² + 2xy + 3z + 4 is 4.

Explanation:

The Laplacian of a function is a scalar quantity that is used to determine the rate of change of the gradient of a function at any point in space. It is a measure of the curvature of the function's surface. In this case, the Laplacian of the function f(x,y,z) = x² + 2xy + 3z + 4 is calculated by finding the second partial derivatives of the function with respect to x, y, and z, and then adding them together. The Laplacian of the function is found to be 4. This means that the curvature of the function's surface is constant throughout space.