Pages: 21

This chapter introduces vectors in and their algebraic operations: addition, subtraction, scalar and cross product. These are next used to obtain various relationships between lines and planes in R3. The chapter focuses on sets and sequences in Rm. It provides an introduction to point set topology, which contains an account on open, closed and compact sets of Rm. The set R of real numbers is the underlying set for all notions (such as sequences and functions) in the one-variable Calculus. The origin corresponds to zero, and all positive real numbers are represented on the right of it, while the negative real numbers are represented on the left of the origin. The set R2 consists of all pairs (a,b) of real numbers a,b. The chapter also describes parallelogram law for adding and subtracting vectors.

Pages: 20

This chapter presents the notion of a function between several variables. It defines domain, codomain, level sets and graph of a function of several variables, and also discusses quadrics and curves. As a rule of thumb, when establishing the domain of definition of a function one has to take into account that division by zero, taking square roots and logarithms of negative numbers are not allowed. Under the mere condition of continuity, the range of a function is called a surface. The fact that all elementary functions, that is, polynomial, rational, trigonometric, exponential and logarithmic functions in several variables are continuous. The vertical line test that determines whether a shape in R2 is the graph of a function works further for functions of two variables. Indeed, if f(x,y) is a function of two variables, then any vertical line must intersect the traces, and thus the whole surface z=f(x,y), in at most one point.

Pages: 19

This chapter discusses the notion of limit for functions of several variables. It first presents the nonexistence of the limit at a point by the Two-path test. Next, criteria for the existence of the limit are investigated in which the Squeeze Principle plays a crucial role. The concept of the limit of a multivariable function is further used to define the continuity and the continuous extension of a several variable function. The chapter explains the main difference between single and multivariable calculus. For functions of one real variable, the existence of the limit at a point is equivalent to the fact that the two sided limits (to the left and to the right at that point) exist and are equal. All elementary functions (polynomials, rational, trigonometric, exponential and logarithmic functions) are continuous on their domain of definition. This means that one can compute their limit at a point by direct substitution.

Pages: 22

This chapter discusses the differentiability of functions of several variables. It introduces first the partial derivatives, the equation of the tangent plane and the normal line to a surface in. The chapter then focuses on the differentiability which is further extended to vector valued functions. Geometrically, the derivative of a single variable function is the slope of the tangent line to its graph. The chapter discusses the geometric interpretation of the partial derivatives of a function. The plane that contains the tangent lines T1 and T2 is called the tangent plane to the surface S at the point P. The line which is orthogonal to the tangent plane at P is called the normal line to the surface S. The chapter also discusses differentiability of Vector-Valued Functions.

Pages: 14

This chapter discusses the Chain Rule for functions of several variables. As a consequence of this result, differentiation of functions defined implicitly is presented. The Mean Value Theorem in several variables setting is also included.

Pages: 14

Let https://www.w3.org/1998/Math/MathML" display="inline"> f : D ⊂ R 2 → R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003449652/afdcc6b9-9be3-4ad4-883c-989795168de8/content/math6_1.tif"/> be a function defined on an open set D containing https://www.w3.org/1998/Math/MathML" display="inline"> ( a , b ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003449652/afdcc6b9-9be3-4ad4-883c-989795168de8/content/math6_2.tif"/> . Then https://www.w3.org/1998/Math/MathML" display="inline"> ∂ f ∂ x ( a , b ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003449652/afdcc6b9-9be3-4ad4-883c-989795168de8/content/math6_3.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> ∂ f ∂ y ( a , b ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003449652/afdcc6b9-9be3-4ad4-883c-989795168de8/content/math6_4.tif"/> measure the rate of change of f in the direction of x and y axis respectively.

Pages: 15

This chapter focuses on second order derivatives of functions of several variables. It discusses the Hessian matrix and its properties. As an application, the Laplace operator is introduced which is a key differential operator in mathematical modelling. The Chain Rule can be extended to second order derivatives as well. One important quantity that involves the second order derivatives is the Laplace operator (or simply Laplacian). The term Laplace operator was coined after the French mathematician Pierre-Simon de Laplace and has since been used in the mathematical modelling of various real life phenomena such as: heat and fluid flow through diffusion equation; electric and gravitational potentials through Poisson's equation; and the wave function in quantum mechanics through Schrödinger's equation. The field of mathematics that studies the harmonic functions is called Potential Theory.

Pages: 12

This chapter introduces higher order derivatives of functions of several variables. These are used to state the Taylor's Approximation Theorem. Further, linear and quadratic approximations are derived which approximate a regular function around a given point as a linear or quadratic polynomial.

Pages: 15

This chapter discusses the Implicit Function Theorem which has been partially used in Chapter 5 when the reader was introduced to implicit differentiation. In the current chapter a rigorous justification of differentiability is provided in the two- and three-variable setting and a general form of the Implicit Function Theorem is stated for any number of variables.

Pages: 16

This chapter discusses the local and global extrema for functions of several variables. The theory is built around the Second Order Derivative Test which allows to classify the critical points into local minima, local maxima and saddle points. In such a context, the second order derivatives and the properties of the Hessian matrix play an important role.

Pages: 25

This chapter introduces the Lagrange Multipliers Method for finding the extrema of functions subject to certain constraints. The method is further applied to various settings such as: finding extrema of functions defined on compact sets, derive various inequalities involving real numbers, modelling problems in business and economics. The Lagrange Multipliers Method can be used to prove some important inequalities in algebra and trigonometry. In Economics, utility functions are used to describe consumers' relative preference for two or more goods or services. One simple example is the preference for certain types of foods, clothes or internet providers. In utility function problems the Lagrange multiplier is called the marginal utility of money.

Pages: 97

Let https://www.w3.org/1998/Math/MathML" display="block"> u = ( u 1 , u 2 , … , u m ) v = ( v 1 , v 2 , … , v m ) w = ( w 1 , w 2 , … , w m ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003449652/afdcc6b9-9be3-4ad4-883c-989795168de8/content/math12_1.tif"/>