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Unformatted text preview: MA1104 Multivariable Calculus Lecture 1 Dr KU Cheng Yeaw Monday Jan 12, 2008 Overview In this lecture, • we introduce coordinate systems for threedimensional space. This provides the setting for our study of calculus of functions of two variables. • we introduce vectors geometrically and study their algebraic properties. We emphasize the power of algebraic manipulation of vectors. 3D Coordinate Systems We can represent a point on a plane by an ordered pair ( a,b ) of real numbers, where a is xcoordinate and b is the ycoordinate. To locate a point in space R 3 , we need three numbers. First, we need to set up a coordinate system as follows: • A fixed point O (the origin) • Three directed lines through O that are perpendicular to each other. They are labeled as xaxis, yaxis and zaxis respectively. This is how we always draw the axes. The direction of the zaxis is determined by the righthand rule Any two of the axes determine a plane. Example 1. Describe and sketch the surface in R 3 represented by the equation x + y = 2 . Solution. x + y = 2 represents a line on the xyplane. However, in R 3 , it represents the plane containing all points whose x and ycoordinate sum to 2 . This is a vertical plane. Distance Formula in 3D The distance  P 1 P 2  between the points P 1 ( x 1 ,y 1 ,z 1 ) and P 2 ( x 2 ,y 2 ,z 2 ) is  P 1 P 2  = p ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 + ( z 2 z 1 ) 2 . Construct a rectangle box as follows: Note that  P 1 P 2  2 =  P 1 B  2 +  BP 2  2 . Clearly,  BP 2  =  z 2 z 1  . By the Pythagorean Theorem,  P 1 B  2 =  P 1 A  2 +  AB  2 , =  x 2 x 1  2 +  y 2 y 1  2 . Combining these equations,  P 1 P 2  2 =  x 2 x 1  2 +  y 2 y 1  2 +  z 2 z 1  2 , as desired. Consequently, we have the following equation of a sphere: Equation of a sphere An equation of a sphere with center C ( h,k,l ) and radius r is ( x h ) 2 + ( y k ) 2 + ( z l ) 2 = r 2 . Vectors Imagine a particle moving in space. Knowing the position of this particle is unsatisfactory as this only gives a static picture of the particle. We need a way to tell us the direction of this moving particle. This can be done by using a vector . A vector is often represented by an arrow. • The length of the arrow represents the magnitude of the vector. • The arrow points in the direction of the vector. For instance, suppose a particle moves along a line segment from point A to point B . The vector v has initial point A (the tail) and terminal point B (the tip). We indicate this by writing v =→ AB . Call this the displacement vector of a particle from A to B . We denote a vector by either: • Printing a letter in boldface v , or • Putting an arrow above the letter ~v ....
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 Spring '08
 Kuchengyeaw
 Multivariable Calculus

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