Single and Multivariable. Calculus. Early Transcendentals .. A few figures in the pdf and print versions of the book are marked with “(AP)” at the end of the. MA Multivariable Calculus. Lecture Notes1. Wong Yan Loi. Department of Mathematics, National University of Singapore, Singapore This is the text for a two-semester multivariable calculus course. The setting The book's aim is to use multivariable calculus to teach mathematics as a blend of.

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Chapter One - Euclidean Three Space. Introduction. Coordinates in Three -Space. Some Geometry. Some More Geometry--Level Sets. Chapter. Multivariable calculus is just calculus which involves more than one variable. To do it properly, you have to use some linear algebra. Otherwise. 14, Non-independent variables, Week 6 summary (PDF). 15, Partial differential equations; review. III. Double integrals and line integrals in the plane. 16, Double .

In the outline that follows, we draw attention to several sections that may be regarded as optional. Chapter numbering follows that of our eight chapter single variable text. Chapter 9, the first chapter of multivariable calculus, begins with a section that introduces vectors in the plane. It is followed by a section that covers the same material in space.

After sections on the dot and cross product, Chapter 9 concludes with a comprehensive account of lines and planes in space.

Tan S.T. Multivariable Calculus

Chapter 10 is devoted to the differential calculus and geometry of space curves. The final two sections are concerned with curvature and associated concepts, including applications to motion and derivations of Kepler's Laws.

Because these two sections are not used later in the text, they may be omitted by instructors who need additional time for other topics. The differential calculus of functions of two and three variables is taken up in Chapter All topics in the standard curriculum are discussed. One less con ventional topic that we treat is the development of order 2 and order 3 Taylor polynomials for functions of two variables.

We use these ideas in the discriminant test for saddle points and local extrema but nowhere else.

It is therefore feasible to omit the discussion of multivariable Taylor polynomials. Chapter 12 is devoted to multiple integrals and their applications. In a section that precedes cylindrical and spherical coordinates, we develop polar coordinates ab initio.

This section may be omitted, of course, if polar coordinates have been introduced earlier in the calculus curriculum. The concluding chapter on vector calculus, Chapter 13, covers vector fields, line and surface integrals, divergence, curl, flux, Green's Theorem, Stokes's The orem, and the Divergence Theorem. Structural Elements We start each chapter with a preview of the topics that will be covered. This short initial discussion gives an overview and provides motivation for the chapter.

Each section of the chapter concludes with three or four Quick Quiz questions located before the exercises.

Chapter 10 Multivariable calculus.pdf - Chap

Most are quick checks of the basic computations of the section. The final section of every chapter is followed by a summary of the important formulas, theorems, definitions, and concepts that have been learned.

This end-of-chapter summary is, in tum, followed by a large collection of review which are similar to the worked examples found in the chapter.

We hope that students and instructors will find these supplementary dis cussions to be enlightening. Which of the graphs below is most likely to represent q as a function of p1 and p2?

C difficulty: Match the following function with the graphs below. Match the function with the graph below. A B Ans: B difficulty: The graph of the function f x, y is shown below. Two contours of the function f x, y corresponding to different values of f cannot ever cross. True difficulty: A False B True Ans: Does this graph support or contradict the claim that the largest annual temperature variations are found on the coasts of continents?

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This graph supports the claim that the largest annual temperature variations are found on the coasts of continents, as level curves are very close together near the coasts of continents. Draw a possible contour diagram for the function whose graph is shown below. Label your contours with reasonable z-values.

Page 14 Full file at https: Functions of Several Variables difficulty: What is your altitude? If you start to move on the surface parallel to the y-axis in the direction of increasing y, does your height increase or decrease? Increase difficulty: The diagram below shows the contour map for a circular island. Sketch the vertical cross-section of the island through the center.

Your sketch should show concavity clearly. Page 15 Full file at https: A B C Page 16 Full file at https: Sketch isothermal curves i. Be sure to label each contour. Page 17 Full file at https: A B Page 18 Full file at https: Functions of Several Variables Ans: The picture below is the contour diagram of f x, y. The areas between contours have been shaded.

Lighter shades represent higher levels, while darker shades represent lower levels. Page 20 Full file at https: Find the contour of f that passes through the point 0, 2.

Below is a contour diagram depicting D, the average fox population density as a function of xE , kilometers east of the western end of England, and xN , kilometers north of the same point. Is D increasing or decreasing at the point , 25 in the northern direction? The function is increasing in the northern direction since as we go north the number of foxes increases. True or False: If all of the contours of a function f x, y are parallel lines, then the function must be linear. False difficulty: A plane passes through the points 1, 3, 7 , -1, 0, 6 , and 2, 1, —3.

Determine the equation of the plane. Consider the plane that passes through the points 1, 3, 10 , -1, —1, 0 , and 2, 1, —1. If you were walking on this plane with no change in altitude, what would be the slope of your path in the xy-plane?

Multivariable Calculus with Applications

Given the table of some values of a linear function complete the table: Given the table of some values of a linear function determine a formula for the function. Find an equation for the plane passing through 1, —5, —2 and containing the x-axis. Find a formula for the linear function whose contours are shown below. Find an equation for f.

Determine the equation of the plane which passes through the point 1, 3, —1 , has slope 5 in the x-direction and slope —3 in the y-direction.

Describe the set of points whose distance from the y-axis equals the distance from the xz-plane. Can the set be described as the graph of a function of two variables? This is two cones centered around the y-axis with vertex at the origin as shown below. If so, what is the equation of that level surface? The level set is a cylinder of radius 4, with its axis along the y-axis.

Some level surfaces of the functions g1 x, y, z , g2 x, y, z and g3 x, y, z are shown below, respectively. Which function takes the value 0 at 0, 0, 0? The level surfaces of the functions g1 x, y, z , g2 x, y, z and g3 x, y, z are shown in below respectively.

Which function is is decreasing in the positive z-direction? Classify the following surface as ellipsoid, elliptical paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets, or cone. There are other possible answers. Consider the figure shown below.

Match the level curve with the function. What value of c if any makes the following function continuous at 0, 0? Describe the level curve of f at level 2. Assume that different contours represent different values of the function.

Explain your answer.

The limit does not exist. The different contours meet at the origin. If it existed the limit would have to be equal to the value on each of the contours.

This is impossible since the values are different. Hence the function is not continuous at the origin. Determine the values of a and b such that the following function is continuous everywhere.

Determine, if possible, the values of the following limits. Note that the limit may not exist nor be determined from the given information. What is the domain of f? What are the possible values of f?

The value of f can be any real number between -1 and 1. There are many possible answers. Simplify the following expression.Calculate the limit of r t as t c. Because equation 9. Solution Using Theorem lb and then la, we have.

By choosing one, say R, we obtain D is a Thus. Woodman pulls with a strength of pounds, then how hard must. This answer agrees with the distance that we calculated in the Insight following Example According to the theory of aerodynamics, the spin causes a difference in air pressure on the sides of the ball and results in a contribution of spin.

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