Section 4.3 - Partial Derivatives

Introduction to Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus, allowing us to analyze how a function of several variables changes as one variable varies while others are held constant. This section introduces the definition, notation, and geometric interpretation of partial derivatives.

• Partial Derivative with respect to x: For a function , the partial derivative with respect to at is defined as:

• Partial Derivative with respect to y: For a function , the partial derivative with respect to at is:

• Notation: and are commonly used, as well as and .

Functions of Two Variables: Partial Derivatives as Functions

For a function , the partial derivatives and are themselves functions of and :

Notation for Partial Derivatives

Let , then:

Rules for Finding Partial Derivatives

To compute partial derivatives:

1. Treat as a constant and differentiate with respect to .

2. Treat as a constant and differentiate with respect to .

Examples of Partial Derivatives

• Example 1: If , find and . Solution:

  Interpretation: These values represent the slopes of the tangent lines to the surface at the point in the and directions, respectively.

• Example 2: If , find and and interpret these numbers as slopes. Solution:

  Interpretation: These values represent the rate of change of in the and directions at .

• Example 3: If , calculate and . Solution:

• Example 4: Find , , and if . Solution: Apply the product and chain rules to compute the derivatives. Additional info: Full expansion and differentiation can be performed for practice.

Geometric Interpretation of Partial Derivatives

Partial derivatives can be interpreted as the slopes of tangent lines to the surface along the and directions. The tangent plane at a point intersects the surface along curves parallel to the and axes, and the slopes of these curves are given by and .

• represents the rate of change of with respect to when is fixed.

• represents the rate of change of with respect to when is fixed.

Higher Order Partial Derivatives

Partial derivatives can be taken multiple times, leading to second and higher order derivatives. For :

These derivatives measure how the rate of change itself changes as you vary each variable.

• Example 5: Find the second partial derivatives of . Solution:

Clairaut's Theorem

Clairaut's Theorem states that if is defined on a disk containing and the mixed partial derivatives and are both continuous on , then:

This theorem guarantees the equality of mixed partial derivatives under suitable continuity conditions.

Partial Differential Equations (PDEs)

An equation involving a function and its partial derivatives is called a partial differential equation (PDE). PDEs are fundamental in physics, engineering, and applied mathematics.

• Laplace's Equation:

• Example 7: Show that is a solution of Laplace's equation. Solution: Compute and and verify their sum is zero. Additional info: Such solutions are called harmonic functions.

• Example 8: Show that is a solution of the wave equation: Solution: Compute the second derivatives and verify the equation holds.

Summary Table: Notation and Derivatives

Additional info: Partial derivatives are essential for understanding gradients, tangent planes, and solving partial differential equations in higher mathematics and applied sciences.