Level Surfaces of Quadratic Functions

Partial Derivatives

• Several variables can interact in a variety of ways
• Derivatives of several variables are more varied and interesting
• Applications are more diverse than single-variable calculus for integrals involving several variables

Functions of Several Variables

• Real-valued functions of several independent real variables are defined analogously to functions of a single variable
• Points in the domain are now ordered n-tuples of real numbers (e.g., (x₁, x₂, ..., xₙ))
• Values in the range are real numbers
• "w" is the dependent variable
• The "xᵢ" are input variables
• The "w" is the output variable

Domains and Ranges

• Functions of several variables have restrictions to prevent complex numbers or division by zero
• The domain of a function is the largest set for which the defining rule generates real numbers, unless otherwise specified
• The range consists of the set of output values for the dependent variable

Graphs, Level Curves, and Contours of Functions of Two Variables

• Level curves of f(x,y) are sets of points in the xy-plane where the function f has a constant value
• The graph of f is also called the surface z = f(x, y)
• A contour curve is the curve in space where a plane z = c cuts a surface z = f(x, y); the curve corresponds to a level curve in the domain of f
• Level surfaces of f(x, y, z) are sets of points (x, y, z) in space where the function f has a constant value f (x, y, z) = c

Functions of Three Variables

• Points where a function of three independent variables has a constant value f(x, y, z) = c form a level surface
• Level surfaces of f(x, y, z) can't be effectively sketched in three-dimensional space

Partial Derivatives

• Partial derivative of f(x, y) with respect to x at (x₀, y₀) is the ordinary derivative of f(x, y₀) with respect to x at x = x₀ (holding y₀ constant)
• Partial derivative of f(x, y) with respect to y at (x₀, y₀) is the ordinary derivative of f(x₀, y) with respect to y at y = y₀ (holding x₀ constant)

Directional Derivatives and Gradient Vectors

• Directional Derivative of f at P₀(x₀, y₀) in the direction of a unit vector u = u₁i + u₂j is the rate of change of f at P₀ in the direction of u
• Gradient vector (∇f) is a vector defined as ∇f = (∂f/∂x)i + (∂f/∂y)j (or ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k in 3D).
• The directional derivative of f at P₀ in the direction of a unit vector u is the dot product of the gradient Vf at P₀ with the vector u: Duf = Vf • u

Tangent Planes and Differentials

• The tangent plane to f(x, y, z)=c at P₀(x₀, y₀, z₀) where the gradient is not zero is the plane through P₀ normal to ∇f|p₀
• The normal line of the surface at P₀ is the line through P₀ parallel to ∇f|p₀

The Chain Rule

• The Chain Rule formula for a differentiable function w = f(x, y) when x = x(t) and y = y(t) are both differentiable functions of t is this: (dw/dt) = fx(x(t), y(t)) * x'(t) + fy(x(t), y(t)) * y'(t)
• Chain Rule in higher dimensions extends (analogous to the single-variable case).
• Diagrams help visualizing multivariable dependencies

Extreme Values and Saddle Points

• Continuous functions of two variables attain extreme values on closed, bounded domains.
• Extreme values occur at either critical points (interior points where both partial derivatives are zero) or boundary points.
• Second Derivative Test classifies critical points as local maxima, minima, or saddle points

Lagrange Multipliers

• Lagrange multipliers provide a method for finding constrained extreme values of a function subject to one or more constraints.
• The method is based on the relationships between gradient vectors
• If a constrained maximum or minimum value exists, the gradient of the function involved is orthogonal to the surface of the constraint

Explore the characteristics of level surfaces represented by the equation $2x^2 + y^2 + z^2 = c$. This quiz examines the behavior of the function as points move away from the origin and the relationship between different level surfaces as the constant $c$ varies. Delve into the implications of setting $c$ to zero and how the function's value changes when moving between points on spherical surfaces.