About Calculus Three

Vector Functions. This topic of calculus introduces vector functions with the assumption the reader is familiar with vectors of three dimensions in the Cartesian coordinate system. First we define a vector-valued function and provide several examples and then we describe how to perform operations with vector-valued functions. Basic properties of vector-valued functions are also detailed.

Limits of Vector Functions. The notion of limits for vector-valued functions is a direct generalization of limits for a real-valued function and again builds upon the notion of distance and proximity. This topic of calculus, illustrates the basics of taking limits and determining intervals of continuity of vector-valued functions.

Derivatives and Integrals of Vector Functions. The notions of derivatives and integrals for vector-valued functions are direct generalizations from real-valued functions and build upon the notions of "slope of a tangent line" (tangent vector) and "rate of change" (velocity). This topic of calculus illustrates the basics of taking derivatives and integrals of vector-valued functions. Smoothness of vector-valued functions is also detailed.

Tangent Vectors and Curvature. A tangent vector points in the direction of motion of a moving body and the normal vector (orthogonal to a tangent vector) points in the direction the body is turning. Using an arc length parametric representation of the path of a moving body, the position, speed, direction, distance, acceleration, and curvature is described in detail. Several formulas are illustrated for the curvature of a vector-valued function and a function of one-variable.

Multivariate Functions. Functions of one variable map a set of real numbers to another set of real numbers. A function assigns, to each number in the first set (domain), a unique real number in the second set (range). A function of several variables has as its domain a set of ordered pairs and its range is a set of real numbers. In general, a function of several variables assigns to each n-tuple a unique real number. For example, a function of two independent variables, assigns to each ordered pair (point in 2-space) a unique real number; and a function of three independent variables assigns to each ordered triple (point in 3-space) a unique real number, etc. In general, the number of coordinate axes needed to graph a function of several variables is the number of variables plus one. Techniques for graphing functions, including domain, range, level curves, and level surfaces are illustrated in detail.

Limits and Continuity of Multivariate Functions. In describing limits of functions of one variable, a variable quantity is approaching a given real number. This approach can be one-sided or two-sided since this process is taking place along only one coordinate axis. This limit will exist if and only if the limiting value is the same approaching from the left side and right side. With functions of two variables, a variable quantity is approaching a given point; and this approach can, in general, have many paths. In order for a limit of a function of two variables to exist the limiting value must be the same, independent of the path taken to the given point. In this topic of calculus, we detail techniques for evaluating limits, epsilon-delta definitions, and continuity.

Partial Differentiation. In general, a derivative is a rate of change or a slope of a tangent line; and the process of differentiating a several-variable function with respect to one independent variable, by using the rules from the calculus of one variable, is called partial differentiation. In this topic of calculus, we illustrate how partial derivatives can be interpreted geometrically as the slopes of the tangent lines at a given point to the traces of a surface in the planes determined by the coordinates of the given point. Higher-order partial differentiation of functions of several variables and Clairaut's Theorem are also detailed.

Differentiable Functions. A function of one variable is differentiable at a point if its derivative has a value at that point. However, the term differentiable is not used the same for functions of several variables; in particular, a function of several variables is not necessarily differentiable if its partial derivatives have values at a given point. With functions of two variables, a function having a tangent plane is equivalent to a function being differentiable, and similarly for functions of several-variables. As is the case for functions of one variable, several-variable functions that are differentiable are also continuous. This topic of calculus, illustrates the sufficient conditions for differentiability; techniques for finding the equation of a tangent plane; and using differentials for approximations.

Chain Rules. The formulas for finding derivatives of a composition of functions are called chain rules. The general case is when the variables of a several-variable (continuous) function are themselves (continuous) functions of several variables. These rules and implicit differentiation are detailed in this topic of calculus.

Directional Derivatives. Directional derivatives are generalizations of partial derivatives. The directional derivative of a several-variable function can be interpreted as the slope of the tangent line for a trace in a given direction. The directional derivative can be concisely expressed in terms of the gradient function. The gradient at a given point yields the largest value of the directional derivative. Indeed the gradient answers the question, in which direction does a function of several variables change fastest and what is the maximum rate of change. The gradient also yields a simple formula for the equation of the tangent plane.

Optimization. The problem of optimizing a real-valued function of several variables is commonplace in all branches of science and mathematics. The notions of relative maximum and minimum are defined and then techniques for finding absolute maxima and minima are detailed including the first and second partials test. In many applied problems, the main focus is on optimizing a function subject to constraint; for example, finding extreme values of a function of several variables where the domain is restricted to a level curve (or surface) of another function of several variables. Lagrange multipliers is a general method which can be used to solve such optimization problems. This topic of calculus, also illustrates the use of Lagrange multipliers for several parameters.

Double Integrals. In this topic of calculus, the distinction between a double integral and an iterated integral in two variables is explained. Indeed, the double integral is often evaluated by converting it to an equivalent iterated integral, which is usually easier to compute; but nonetheless double integrals and iterated integration are distinct concepts. Essentially, an iterated integral is like the inverse of mixed partial differentiation and a double integral is a direct extension of the Riemann integral (Riemann sums) of one variable to functions of two independent variables.

Surface Area. In this topic of calculus, we apply double integrals to the problem of computing the area of a surface over a region. We demonstrate a formula that is analogous to the formula for finding the arc length of a one variable function and detail how to evaluate a double integral to compute the surface area of the graph of a differentiable function of two variables; as well as surfaces with a parametric representation.

Triple Integrals. First, triple integrals are defined for boxed regions and their generalization from the Riemann integral of a one and two variable function is described in detail. The relationship between triple integrals and iterated integrals (Fubini's Theorem) is detailed. The distinction between a triple integral and an iterated integral in three variables is explained. Indeed, the triple integral is often evaluated by converting it to an equivalent iterated integral, which is usually easier to compute; but nonetheless triple integrals and iterated integration are distinct concepts. Essentially, an iterated integral is like the inverse of mixed partial differentiation and a triple integral is a direct extension of the Riemann integral (Riemann sums) of one and two variable to functions of three independent variables.

Applications of Multiple Integrals. Applications of multiple integrals including planar lamina, mass, moments and probability density functions are detailed in this topic of calculus.

Change of Variables and Jacobians. For functions of one variable the process of changing variables introduces a factor into the integrand and the same is true for the multiple variable cases. In general, the factor is called the Jacobian of the change of variables and it depends on a one-to-one and onto transformation between the two regions of integration. The Jacobian of the transformation must be nonzero, and we will assume that the original variables are continuously differentiable functions of the new variables over the domain of the transformation. Examples of these types of transformations are mappings from the rectangular to polar coordinates, rectangular to cylindrical coordinates, and rectangular to spherical coordinates. Examples of other change of variables are also illustrated, including a discussion on how to choose your own change of variables.

Vector Fields. Vector fields can be used to quantify the amount of work done by variable force acting on a moving body. Measuring the amount of force (fluid flow, electric charge, etc.) can sometimes be achieved by computing an integral of a vector field with respect to an orientable curve or surface. By and large, a vector field is a function that assigns a vector to each point in a set of ordered pairs. Notice that it may be impossible to draw every vector in a vector field (drawing a vector at every point in space); even so, drawing a few key representatives (vectors) usually will give a reasonable impression of a vector field. Common examples of vector fields include force fields, velocity fields, gravitational fields, magnetic fields, and electric fields. This topic of calculus, details techniques of drawing vector fields and illustrates properties of the div, curl, and Laplacian, which are used in computing integrals of vector fields. Finally, conservative vector fields are defined and finding the scalar potential function is illustrated.

Line Integrals. Line integrals have a great many applications in the sciences, but were originally constructed mainly to compute work and mass in physics. Generally, if direction can be described along a curve using a parametric representation (called an orientable curve), then as a generalization of the Riemann integral it is possible to define a line integral where Riemann sums are constructed along the piecewise-smooth orientable curve. This topic illustrates how line integrals can be evaluated by reducing the line integral into a familiar one-variable integral with respect to the variable of parameterization of the piecewise-smooth orientable curve. Techniques for evaluating line integrals and applications are also detailed.

Path Independence. In general, the value of a line integral depends on the path of integration, but in certain cases, the line integral will be the same for all paths in a given region with the same initial point and terminal point; and if so the line integral is called independent of path. Additionally, a vector field that is the gradient field for a scalar function on some region is called a conservative vector field on that region. So now recall, for functions of two variables, the analogue of the derivative is the gradient, and the corresponding analogue of the fundamental theorem of calculus is the fundamental theorem for line integrals, which to our advantage, offers us a way to evaluate a line integral of a conservative vector field by simply knowing the values of the scalar (potential) function at the endpoints of the path. This topic of calculus illustrates several techniques for evaluating a line integral and details how the line integral of any conservative vector field is independent of path (and thus is zero for closed paths) and that these are the only such vector fields that are independent of path.

Green's Theorem. Green's theorem elegantly yields another technique to evaluate a line integral of a vector field and is especially important when the vector field is not conservative. To do so, Green's theorem assumes that we have a simply connected region that is bounded by a positively oriented piecewise smooth Jordan curve. In this case, Green's theorem offers us a way to evaluate a line integral of a continuously differentiable vector field by computing a double integral involving the partials of the component functions of the vector field. It can be thought of as a generalization of the fundamental theorem of calculus for definite integrals because it gives us a way to evaluate a double integral using only the values of partial antiderivatives on the boundary of the domain of integration. Green's theorem for doubly connected regions and alternate forms of Green's theorem involving the curl and div of a vector field are also detailed.

Cite this as:
About Calculus Three
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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