Slide_16 - MA1104 Multivariable Calculus Lecture 16 Dr KU...

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Unformatted text preview: MA1104 Multivariable Calculus Lecture 16 Dr KU Cheng Yeaw Thursday March 19, 2009 Recap Last Lecture ... • We introduced vector fields. An important examples is gradient vector field. We say that F is a conservative vector field if F is the gradient vector field of some scalar function f , i.e. F = O f . • We defined line integral of a scalar function along a (piecewise) smooth curve C with respect to arclength, x and y respectively. It generalizes the usual single integral. Overview In this lecture, • We introduce another type of line integral. It is an integral of a vector field (instead of scalar field) along a curve. This type of line integrals appear frequently in physics. • It turns out that the computation of this line integral simplifies dramatically (in particular, it depends only on the initial and terminal points of a curve) provided the vector field is conservative. This is the Fundamental Theorem for Line Integrals. • We want to answer the question how to determine whether a vector field is conservative. Work Done by Force Field Lets start with the following practical problem in physics: Compute the work done on an object by a force field F ( x,y,z ) as the object moves along a curve C . Recall that in the special case when F is constant in moving the object from a point P to Q (i.e. the curve C is just a straight line), we have Work done = F · D where D =--→ PQ is the displacement vector. We shall suppose that F ( x,yz ) = P ( x,y,z ) i + Q ( x,y,z ) j + R ( x,y,z ) k is continuous on R 3 . We wish to compute the work done by this force in moving a particle along a smooth curve C . Step 1. We divide C into subarcs P i- 1 P i with lengths 4 s i by dividing the parameter interval [ a,b ] into subintervals of equal width. Figure for two-dimensional case: Figure for three-dimensional case: Step 2. Choose a point P * i ( x * i ,y * i ,z * i ) on the i-th subarc corresponding to the parameter value t * i . If 4 s i is small, then as the particle moves from P i- 1 to P i along the curve, it proceeds approximately in the direction of T ( t * i ) , the unit tangent vector at P * i . Step 3. Calculating work done. Thus, the work done by the force F in moving the particle P i- 1 from to P i is approximately F ( x * i ,y * i ,z * i ) · 4 s i T ( t * i ) = ( F ( x * i ,y * i ,z * i ) · T ( t * i )) 4 s i . The total work done in moving the particle along C is approximately n X i ( F ( x * i ,y * i ,z * i ) · T ( x * i ,y * i ,z * i )) 4 s i where T ( x,y,z ) is the unit tangent vector at the point ( x,y,z ) on C . Intuitively, we see that these approximations ought to become better as n becomes larger. Thus, we define the work W done by the force field F as the limit of the Riemann sums in the preceding formula, namely, W = Z C F ( x,y,x ) · T ( x,y,z ) ds = Z C F · T ds....
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Slide_16 - MA1104 Multivariable Calculus Lecture 16 Dr KU...

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