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Tutorial 9 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 9 (Solution Notes) 1. A table of values of a function f is given below. . . . . . . . . . . 2 8 2 9 1 3 5 7 0 1 6 4 0 1 2 x y For example, f (1 , 1) = 5. Find C f d r , where (a) C has parametric equations x = t 2 + 1 , y = t 3 + t, 0 t 1 . (b) C is the unit circle x 2 + y 2 = 1. Preliminaries Recall: Fundamental Theorem of Calculus If f is continuous at every point of [ A, B ], then B A f ( x ) dx = f ( B ) f ( A ) . Remark: 1. This is the second type of Fundamental Theorem of Calculus, and the other type is equivalent to the above one. 2. For multi-variable functions, the gradient f is composed of partial derivatives: f ( x, y ) = f x f y , f ( x, y, z ) = f x f y f z , · · · Therefore, if we replace f with f in the Fundamental Theorem for multi-variable function f , we can get the general form of Fundamental Theorem: (Of course, here the integral is along lines in vector spaces, so line integrals!) Fundamental Theorem of Line Integral If the integral is independent of the path from A to B , its value is B A f d r = f ( B ) f ( A ) . 177
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Remark: 1. Here the path can be any curve: straight line, curve in a plane, curve in vector space. 2. The single-variable integration is a special case of line integration, that is do the integration along x -axis. 3. The Fundamental theorem says that the value of a line integral is just depending on the limit points of the integral. Such functions (equal to some f ) are called conservative fields in Physics. For example, if there is no external force (for example, no friction force here), then the total energy (including kinetic energy and potential energy) of an object is conserved when moving from A to B . Recall: Field In Physics, the notation field stands for a multi-variable function. So, we have learnt two kinds of fields: scalar field and vector field. Check tutorial notes 8 for more details. Recall: Conservative Field We have many ways to define the conservative field, and all the definitions are equivalent. Here is the way used in Thomas’ Calculus: DEFINITIONS Path Independence, Conservative Field Let F be a field defined on an open region D in space, and suppose that for any two points A and B in D the work done in moving from A to B is the same over all paths from A to B . Then the integral is path independent in D and the field F is conservative on D . 1 F # d r 1 B A F # d r An alternative definition is: A field F is conservative if and only if it is the gradient field of a scalar function f ; that is, if and only if for F = f for some f . The function f has a special name: potential function . 178
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Remark: 1. Please note that the conservative function is F , not the potential function f . 2. Revisit tutorial 8 to check the criteria of conservative fields (Component Test), and the method to get the potential function f .
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