vector cal. please show step where is a Cl function. Let it be defined in the wh

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vector cal. please show step

where is a Cl function. Let it be defined in the whole plane. We are looking for functions y(x) defined by this equations, that is, functions y(r), r e that f(x,y()) 0 for any x in A. the case of a single equation in two variables f(x, y)0 (a, b) such In general we can not say if there are such functions v(x), how many such func- tion may exist and what may be their domains (correspondi existence theoren a solution y Vo of the equation f(ra) for v. We also have to assume that yiro, 30) 0. Then we can assert that there is an interval containing ro and an function y(r), zeA (a,b) such that f(x, y(x)) 0 for any r in A (implicit function determined by equation f(z, y) interval . 0. Theorem does not say anything about the size of There is a a geometric approach to the problem. If we can produce a good sketch of the curve in plane C defined by equation f(x, y) 0 then solutions of the equations for y with a fixed value x = za are obtained as intersections of C with vertical lines rTo. In this context we have the following Problem 1. Sketch the curve C defined by the equatio and use it to describe (characterize) implicit function(s) v/(x) determined by this equa- tion. In particular, determine the intervals A where whose implicit functions are de- fined

Explanation / Answer

In plane : A point in a region in the plane is an interior point of if it is the center of a disk that lies entirely in . A point is a boundary point of if every disk centered at contains points that lie outside as well as points that lie in (The boundary point itself need not belong to ) . The interior of the region is the set of interior points of The boundary of the region is the set of boundary points of A region is open if it consists entirely of interior points. A region is closed if it contains all of its boundary points.

By a parametrized curve, or simply a curve, in R n , we mean the image C of a continuous function : [r, s] R n , where [r, s] is a closed interval in R. C is called a plane curve, resp. a space curve, if n = 2, resp. n = 3. An example of the former, resp.the latter, is the cycloid, resp. the right circular helix, parametrized by : [0, 2] R 2 with (t) = (t sin t, 1 cost), resp. : [3, 6] R 3 with (t) = (cost,sin t, t). Note that a parametrized curve C has an orientation, i.e., a direction; it starts at P = (r), moves along as t increases from r, and ends at Q = (s). We call the direction from P to Q positive and the one from Q to P negative. It is customary to say that C is a closed curve if P = Q. We say that C is differentiable iff is a differentiable function. Definition. Let C be a differentiable curve in R n parametrized by an as above. Let a = (t0), with t0 (r, s). Then 0 (t0) is called the tangent vector to C at a (in the positive direction). For example, consider the unit circle C in the plane given by : [0, 2] R 2 , t (cost,sin t). Clearly is differentiable, and we have 0 (t) = ( sin t, cost) for all t (0, 2). Consider the point a = (1/2, 3/2) on the circle; clearly, a = (/3). By the above definition, the tangent vector to C at a is given by 0 (/3) = ( 3/2, 1/2). Let us now check that this agrees with our notion of a tangent from one variable Calculus. To this end we want to think of C as the graph of a one variable function, but this is not possible for all of C. However, we can express the upper and the lower semi-circles as graphs of such functions. And since the point of interest a lies on the upper semi-circle C +, we look at the function y = f(x) with f(x) = 1 x 2 , whose graph is C +. Note that (by the chain rule) f 0 (x) = x(1 x 2 ) 1/2 . The slope of the tangent to C + at a is then given by f 0 (1/2) = 1/ 3. Hence the tangent line to C at a is the line L passing through (1/2, 3/2) of slope 1/ 3. This is the unique line passing through a in the direction of the tangent vector ( 3/2, 1/2). (In other words L is the unique line passing through a and parallel to the line joining the origin to ( 3/2, 1/2).) The situation is similar for any plane curve which is the graph of a one-variable function f, i.e. of the form (t) = (t, f(t)). So the definition of a tangent vector above is very reasonable.

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