MthSc 206 Summer1 13 Goddard


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1 16.1 Vector Fields A vector field is a function that assigns to each point a vector. A vector field might represent the wind velocity at each point. For example, F(x, y) = y i+x j represents spinning around the origin. The gradient f is a vector field. A gradient vector field is orthogonal to its contour map. A conservative vector field is one that is the gradient of some (scalar) function (sometimes called a potential function).
2 16.2 Line Integrals The line integral of a scalar function f on some curve is defined as f(x, y, z) ds = b a f (x(t), y(t), z(t)) ( dx dt ) 2 ( ) 2 + dy ( dt + dz ) 2 dt dt for some parametrization x(t), y(t), z(t) of. (Note that if f is always 1 then we get the arc length of.) The line integral with respect to x is defined as f(x, y, z) dx = b a f (x(t), y(t), z(t)) x (t) dt Let be a smooth curve given by r(t) for a t b. The line integral of a vector field F along is the work W done in moving along : W = F dr = b a F(r(t)) r (t) dt For example, W is negative if against the field and positive if with the field. Note that if F = P i + Qj + Rk, then F dr = P dx + Q dy + R dz
3 16.3 Fundamental Theorem for Line Integrals The Fundamental Theorem for Line Integrals says that the value of the line integral of the gradient of a function f is given by the difference in the f values at the ends of the curve (provided f is continuous on ). That is, if = r(t) with a t b, then f dr = f(r(b)) f(r(a)) An open region D is one that contains a disc around every point (that is, D does not contain any of its boundary). A connected region D is one where we can get between any two points of D while staying in D. It is then simply connected if every closed curve in D encircles only points of D. In an open simplyconnected region, F = P i + Q j is conservative P y = Q x F dr is independent of path To find f from F = P i+q j if it exists, start by noting that f = P dx+g(y)
4 16.4 Green s Theorem Green s Theorem says that a line integral over a closed curve equals a double integral over the region bounded by the curve. A curve is piecewisesmooth if it is made up of smooth pieces; a curve is closed if it finishes where it starts; a closed curve is positivelyoriented if it is traversed counterclockwise. Green s Theorem: curve, then If is a positivelyoriented piecewisesmooth closed P dx + Q dy = D Q x P y da The line integral is sometimes written to emphasize that is closed.
5 16.5 url and Divergence We remember curl by the rule curl F = F That is, if F = P i + Q j + R k, then ( R curl F = y Q ) ( P i + z z R ) ( Q j + x x P ) k y Note that curl ( f) = 0 If curl F = 0 everywhere and F is wellbehaved, then F must be a conservative vector field. url is related to how an object placed in the vector field would rotate: if curl F = 0 at some point, F is said to be irrotational at that point. We remember divergence by the rule That is, if F = P i + Q j + R k, then div F = F div F = P x + Q y + R z Note that div (curl F) = 0 Divergence is related to the expansion/contraction of the vector field: if div F = 0 everywhere, F is said to be incompressible.
6 16.6 Parametric Surfaces and their Areas A parametric surface is given by a vector function of two parameters (for example u and v). So in general a surface can be represented by r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k (u, v) D for some domain D. The grid curves are obtained by keeping one of the parameters constant. Example surfaces: (1) The graph of function z = f(x, y) is a surface with x = u, y = v, z = f(u, v). (2) A plane through point r 0 containing (nonparallel) vectors a and b is given by r(t) = r 0 + ua + vb. (3) A sphere is best expressed in spherical coordinates. (4) If you rotate the curve y = f(x) around the xaxis, you get the surface of revolution with parametric equation x = x, y = f(x) cos θ, z = f(x) sin θ. If a parametrization covers the surface S only once, then the area of S is given by A(S) = D r u r v da where r u = x u, y u, z u and r v = x v, y v, z v are called the partial grads. For example, if S is the graph of function z = f(x, y), then r u r v = fx 2 + fy So that we get the result of section 15.6: A(S) = D 1 + f 2 x + f 2 y da
7 16.7 Surface Integrals The surface integral of scalar function f over surface S with parametrization r(u, v) is defined as a double integral: f(x, y, z) ds = f(r(u, v)) r u r v da S D where D is the shadow of S. An orientation of a surface means choosing a positive direction for the normal n at each point such that this choice is continuous. For a closed curve, the convention is that the positive orientation is outward. The surface integral (or flux) of vector function F over oriented surface S with parametrization r(u, v) is F ds = F (r u r v ) da S S
8 16.8 Stokes Theorem Stokes Theorem relates a surface integral to a line integral on its boundary: F dr = curl F ds S where S is an oriented surface with as its positive boundary curve, and F is a vector field.
9 16.9 The Divergence Theorem The divergence theorem relates a volume integral to a surface integral on its boundary: F ds = div F dv S E where E is a simple solid region with S as its oriented (outwards positive) surface.
10 16.10 hapter Summary Integral Double Green Line Definition Stokes Triple Divergence Surface
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