Partial Derivatives in Multivariable Calculus

Partial Derivatives

• Let u = f(x, y) be a function of two variables x and y. If y is constant and x varies, u is a function of x only.
• The derivative of u with respect to x, treating y as constant, is the partial derivative of u with respect to x, denoted as ∂u/∂x or ux.
• Similarily, the partial derivative of u with respect to y (treating x as constant), is denoted as ∂u/∂y or uy.
• These partial derivatives also depend on both x and y so can further be differentiated partially.

Geometric Interpretation

• The slope of the curve z = f(x, y) at a point P(x₀, y₀, f(x₀, y₀)) in the plane y = y₀ is the partial derivative of f with respect to x at (x₀, y₀).

Partial Derivatives of Composite Functions

• If u = f(x, y) where x = g(s, t) and y = h(s, t), then ∂u/∂s and ∂u/∂t can be calculated using the chain rule.

Jacobians

• If u and v are functions of two independent variables x and y, the Jacobian of u and v with respect to x and y is denoted as ∂(u, v)/∂(x, y).
• It's a determinant: ∂(u, v)/∂(x, y) = | (∂u/∂x) (∂v/∂y) - (∂u/∂y) (∂v/∂x) |
• The Jacobian has properties useful in various applications (function dependency, transformations).

Taylor's Series for Functions of Two Variables

• The Taylor series expansion for a function f(x, y) about a point (a, b) includes terms involving partial derivatives.

Maclaurin's Series for Functions of Two Variables

• The Maclaurin series expansion for a function f(x, y) about the point (0, 0) involves partial derivatives.

Maxima and Minima of Functions of Two Variables

• A function z = f(x, y) has a maximum or minimum at (a, b) if f(a + h, b + k) < or > f(a, b) for all small h and k.
• Necessary conditions for a maximum or minimum are fx(a, b) = 0 and fy(a, b) = 0.
• Second-order partial derivatives (r, s, t) help determine if a point is a maximum, minimum, or saddle point.

Errors and Approximations

• Error analysis in calculations involving functions of several variables considers the absolute, relative, and percentage errors for input variables leading to errors in computed results.

This quiz explores the concepts of partial derivatives, their geometric interpretation, and their applications in composite functions. You'll learn about the notation used for partial derivatives and the role of Jacobians in multivariable functions. Test your understanding of these essential calculus topics!