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Notes-C

# Notes-C - Calculus in several variables Gabriel Nagy 0.1...

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Calculus in several variables Gabriel Nagy July 10, 2006

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0.1 General introduction The main objects of calculus are functions and a central idea is that of limit. This idea used on appropriate functions allows to compute both pointwise rate of change of functions and areas of arbitrary regions in the plane. The former is called differential calculus and the latter integral calculus. The fact that differ- ential and integral calculus are deeply related is summarized in the fundamental theorem of calculus. One way to study a certain subject is to start with a simple case and later on concentrate on more complicated situations. A first course in calculus usually focuses on single variable functions, that is, f : R R , denoted as f ( x ). Such functions are simple to represent graphically in the plane. y y = f(x) x y y = f(x) a c b x f (c) The graphical interpretation of both the derivative at c and the integral in [ a, b ] of f ( x ) are also simple to obtain. In the first case f 0 ( c ) is the slope of the line tangent of the graph of f ( x ) at c , while in the second case R b a f ( x ) dx is the area of the shaded region in the picture above. There are several generalizations of these ideas. One possibility is to consider scalar functions of two and three variables, that is, f : R 2 R , denoted as f ( x, y ), and f : R 3 R , denoted as f ( x, y, z ), respectively. A different possi- bility is to consider vector-valued functions, r : R R 3 , denoted as r ( t ). One reason to consider such generalizations is that functions of both types appear frequently in any mathematical description of nature. The latter can be repre- sented by curves is space which are appropriate to describe the motion of point particles. ii
z r(t) r(0) y x z f(x,y) = x + y x y 2 2 The former generalization, functions f ( x, y ), is useful to describe situations like the temperature on the surface of a table. Here ( x, y ) label the points of the table and f is the temperature. Such functions can be represented graphically as above, in the case of f ( x, y ) = x 2 + y 2 . In vector calculus one generalizes the concept of derivative and integral to func- tions of several variables and to vector-valued functions. This generalization requires to introduce in space the ideas of coordinate systems, vectors, lines and planes. This presentation will occupy the first quarter of the course. the rest will be dedicated to generalize the ideas of derivative and integral first to vector-valued functions, and then to scalar functions of two and three variables. iii

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Contents 0.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . ii 1 Vectors in space 1 1.1 Cartesian coordinate systems . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Distance formula . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vectors on the plane and in space . . . . . . . . . . . . . . . . . . 5 1.2.1 Components of a vector . . . . . . . . . . . . . . . . . . . 6 1.2.2 Operations with vectors . . . . . . . . . . . . . . . . . . . 7 1.3 Dot product and projections . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Dot product in components . . . . . . . . . . . . . . . . . 11 1.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 The cross product in components . . . . . . . . . . . . . . 14 1.4.2 The triple product of three vectors . . . . . . . . . . . . . 15 1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Lines and planes in space . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Lines in space . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 Planes in space . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.3 Distance formula from a point to a plane . . . . . . . . . 18 1.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2

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