SOLUTION OF LAPLACE EQUATION IN CARTESIAN COORDINATES || MATHEMATICAL PHYSICS || WITH EXAM NOTES ||

Solution of Laplace Equation in 3D Cartesian Coordinates: Summary

Key Concepts:

• Laplace Equation
• Cartesian Coordinates (x, y, z)
• Method of Separation of Variables
• Partial Differential Equations
• General Solution
• Hyperbolic Functions

1. Introduction and Laplace Equation:

• The video explains how to find the solution of Laplace's equation in 3D Cartesian coordinates (x, y, z).
• Laplace's equation is given by: ∇²V = 0, where V is a scalar function.
• The del-squared operator (∇²) needs to be expressed in Cartesian coordinates.

2. Laplace Equation in Cartesian Coordinates:

• In Cartesian coordinates, the del-squared operator is: ∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z².
• Therefore, Laplace's equation in Cartesian coordinates becomes: ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0.
• The goal is to find the function V(x, y, z) that satisfies this equation.

3. Method of Separation of Variables:

• To solve the equation, the method of separation of variables is used.
• Assume that V(x, y, z) can be written as a product of three functions, each depending on only one variable: V(x, y, z) = X(x)Y(y)Z(z).
• This assumption simplifies the partial differential equation into three ordinary differential equations.

4. Substituting and Separating Variables:

• Substitute V = XYZ into the Laplace equation:
  • Y(y)Z(z) ∂²X(x)/∂x² + X(x)Z(z) ∂²Y(y)/∂y² + X(x)Y(y) ∂²Z(z)/∂z² = 0
• Divide the entire equation by XYZ:
  • (1/X) ∂²X/∂x² + (1/Y) ∂²Y/∂y² + (1/Z) ∂²Z/∂z² = 0
• Each term is now a function of only one variable (x, y, or z).

5. Introducing Separation Constants:

• Since the sum of these three terms is zero, each term must be equal to a constant.
• Let:
  • (1/X) ∂²X/∂x² = -m²
  • (1/Y) ∂²Y/∂y² = -n²
  • (1/Z) ∂²Z/∂z² = p²
• The constants are chosen such that -m² - n² + p² = 0, which implies p² = m² + n².

6. Solving the Ordinary Differential Equations:

• Now we have three ordinary differential equations:
  • ∂²X/∂x² + m²X = 0
  • ∂²Y/∂y² + n²Y = 0
  • ∂²Z/∂z² - p²Z = 0
• These are standard second-order differential equations with known solutions.

7. General Solutions for X, Y, and Z:

• The general solutions are:
  • X(x) = C₁ cos(mx) + C₁' sin(mx)
  • Y(y) = C₂ cos(ny) + C₂' sin(ny)
  • Z(z) = C₃ cosh(pz) + C₃' sinh(pz)
  • Where C₁, C₁', C₂, C₂', C₃, and C₃' are arbitrary constants.
  • cosh and sinh are hyperbolic cosine and sine functions, respectively.

8. Combining the Solutions:

• The general solution for V(x, y, z) is the product of these three solutions:
  • V(x, y, z) = [C₁ cos(mx) + C₁' sin(mx)] * [C₂ cos(ny) + C₂' sin(ny)] * [C₃ cosh(pz) + C₃' sinh(pz)]

9. Relation between Constants:

• The constants m, n, and p are related by the equation: p² = m² + n², or p = √(m² + n²).

10. Conclusion:

• The video provides a step-by-step method to solve Laplace's equation in 3D Cartesian coordinates using the method of separation of variables.
• The final solution is a product of trigonometric and hyperbolic functions, with constants determined by boundary conditions.
• The relationship between the separation constants (m, n, p) is crucial for the solution to be valid.