Mathematics Foundations/11.1 Functions of Several Variables

11.1 Functions of Several Variables 11.1.1 Introduction to Functions of Several Variables A function of several variables maps points from a subset of to . In this section, we primarily focus on scalar-valued functions (where ), which assign a single real number to each point in the domain. 11.1.1.1 Definition A function of variables is a rule that assigns to each ordered -tuple in the domain exactly one real number, denoted by . For , we have functions of two variables, often written as or . For , we have functions of three variables, often written as or . 11.1.2 Domain and Range The domain of a function of several variables is the set of all points in for which the function is defined. The range is the set of all possible output values. Unless otherwise specified, the domain of a function is assumed to be the largest possible subset of for which the function is well-defined. 11.1.2.1 Examples 1. For , the domain is , which is a closed disk of radius 1 centered at the origin. 2. For , the domain is , which is the plane with the line removed. 11.1.3 Graphs and Level Curves/Surfaces 11.1.3.1 Graph of a Function of Two Variables The graph of a function is the set of all points in such that and is in the domain of . This forms a surface in three-dimensional space. 11.1.3.2 Level Curves Level curves (or contour lines) of a function are the curves in the -plane defined by the equation for some constant . These curves connect points in the domain where the function takes the same value. A contour map or contour plot consists of several level curves corresponding to different values of . 11.1.3.3 Level Surfaces For a function of three variables, level surfaces are defined by the equation for some constant . These are surfaces in three-dimensional space where the function takes the same value. 11.1.4 Limits and Continuity 11.1.4.1 Limit of a Function of Several Variables The limit of a function as approaches is , written as: if for every , there exists a such that: whenever 11.1.4.2 Approaching a Point Along Different Paths For functions of multiple variables, limits can be approached along different paths. A limit exists only if the function approaches the same value regardless of the path taken toward the point. For example, for at : - Along the x-axis (), - Along the y-axis (), - Along the line , Since the limit depends on the path of approach, the limit does not exist. 11.1.4.3 Continuity A function is continuous at a point in its domain if: A function is continuous on a region if it is continuous at every point in that region. 11.1.5 Partial Derivatives 11.1.5.1 Definition The partial derivative of a function with respect to , denoted by or , is the derivative of with respect to while holding all other variables constant. For a function of two variables: 11.1.5.2 Geometric Interpretation For a function : - represents the slope of the curve formed by the intersection of the surface and the vertical plane at the point . - represents the slope of the curve formed by the intersection of the surface and the vertical plane at the point . 11.1.5.3 Higher-Order Partial Derivatives Higher-order partial derivatives are obtained by taking partial derivatives of partial derivatives: - Second-order partial derivatives: , - Mixed partial derivatives: , If the mixed partial derivatives are continuous, then they are equal: (Clairaut's theorem). 11.1.6 Differentiability 11.1.6.1 Total Differential The total differential of a function is: For a function of variables: 11.1.6.2 Differentiability A function is differentiable at a point if both partial derivatives and exist and: If a function has continuous partial derivatives in a neighborhood of a point, then it is differentiable at that point. 11.1.7 Tangent Planes and Normal Lines 11.1.7.1 Tangent Plane The equation of the tangent plane to the surface at the point is: 11.1.7.2 Normal Line The normal line to the surface at the point has the direction vector: The parametric equations of the normal line are: 11.1.8 Chain Rule for Functions of Several Variables 11.1.8.1 Single-Variable Case If where and , then: 11.1.8.2 Multi-Variable Case If where and , then: 11.1.9 Directional Derivatives The directional derivative of a function at a point in the direction of a unit vector is: More generally, for a function and a unit vector : 11.1.10 Gradient Vector The gradient of a function is the vector: For a function of two variables: 11.1.10.1 Properties of the Gradient 1. The gradient vector points in the direction of steepest ascent of at . 2. The magnitude of the gradient, , gives the maximum rate of change of at . 3. The gradient is perpendicular to the level curves (or level surfaces) of . 4. The directional derivative in the direction of a unit vector is . 11.1.11 Exercises 1. Find the domain and range of the function . 2. Sketch the level curves of the function for . 3. Calculate the partial derivatives and for . 4. Determine whether the limit exists. 5. Find the equation of the tangent plane to the surface at the point . 6. Compute the directional derivative of at the point in the direction of the vector . 7. Find the gradient vector of and determine the direction of steepest ascent at the point .