Tangent line calculator
#Tangent line calculator free
Vector fields that are both conservative and source free are important vector fields. (Stream functions play the same role for source-free fields that potential functions play for conservative fields.)įind a stream function for vector field F ( x, y ) = 〈 x sin y, cos y 〉. ∇ g ( a, b ) = 0 for any point ( a, b ) ( a, b ) in the domain of g.Since the gradient of g is perpendicular to the level curve of g at ( a, b ), ( a, b ), stream function g has the property F ( a, b ) Geometrically, F ( a, b ) F ( a, b ) is tangential to the level curve of g at ( a, b ). A stream function for F = 〈 P, Q 〉 F = 〈 P, Q 〉 is a function g such that P = g y P = g y and Q = − g x. There is a stream function g ( x, y ) g ( x, y ) for F.In other words, flux is independent of path. If C 1 C 1 and C 2 C 2 are curves in the domain of F with the same starting points and endpoints, then ∫ C 1 F.N d s across any closed curve C is zero.The following statements are all equivalent ways of defining a source-free field F = 〈 P, Q 〉 F = 〈 P, Q 〉 on a simply connected domain (note the similarities with properties of conservative vector fields): If we replace “circulation of F” with “flux of F,” then we get a definition of a source-free vector field. In fact, if the domain of F is simply connected, then F is conservative if and only if the circulation of F around any closed curve is zero. Recall that if vector field F is conservative, then F does no work around closed curves-that is, the circulation of F around a closed curve is zero. ∫ C v ⋅ N ds = ∬ D ( P x + Q y ) d A = ∬ D 8 d A = 8 ( area of D ) = 80. Therefore, by the same logic as in Example 6.40, Let D be any region with a boundary that is a simple closed curve C oriented counterclockwise. The logic of the previous example can be extended to derive a formula for the area of any region D. In Example 6.40, we used vector field F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 to find the area of any ellipse. Therefore, the area of the ellipse is π a b. d r = 1 2 ∫ C − y d x + x d y = 1 2 ∫ 0 2 π − b sin t ( − a sin t ) + a ( cos t ) b cos t d t = 1 2 ∫ 0 2 π a b cos 2 t + a b sin 2 t d t = 1 2 ∫ 0 2 π a b d t = π a b.r 4 ( t ) d t = ∫ a b P ( t, c ) d t + ∫ c d Q ( b, t ) d t − ∫ a b P ( t, d ) d t − ∫ c d Q ( a, t ) d t = ∫ a b ( P ( t, c ) − P ( t, d ) ) d t + ∫ c d ( Q ( b, t ) − Q ( a, t ) ) d t = − ∫ a b ( P ( t, d ) − P ( t, c ) ) d t + ∫ c d ( Q ( b, t ) − Q ( a, t ) ) d t.
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Therefore, the circulation of a vector field along a simple closed curve can be transformed into a double integral and vice versa. This form of the theorem relates the vector line integral over a simple, closed plane curve C to a double integral over the region enclosed by C. The first form of Green’s theorem that we examine is the circulation form.
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In particular, Green’s theorem connects a double integral over region D to a line integral around the boundary of D. Green’s theorem also says we can calculate a line integral over a simple closed curve C based solely on information about the region that C encloses. Green’s theorem says that we can calculate a double integral over region D based solely on information about the boundary of D. Green’s theorem takes this idea and extends it to calculating double integrals. Figure 6.32 The Fundamental Theorem of Calculus says that the integral over line segment depends only on the values of the antiderivative at the endpoints of.