 | Term | Definition |
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Inverse Square Field |
<F(x,y,z)> = k<u>/||<r>²|| |
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Conservative Vector Field |
A vector field where there exist a differential function ƒ such that <F> = ∇ƒ. The function ƒ is called the potential function for <F> |
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Test for conservative vector field in the plane |
Let M and N have continuous first partial derivative on an open disc R. The vector field is given by <F(x,y,)> = M<i> + N<j> is conservative if and only if ∂N/∂x = ∂M∂y |
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Curl of a Vector Field |
∇ x <F> = (∂P/∂y - ∂N/∂z)<i> - (∂P/∂x - ∂M/∂z)<j> +(∂N/∂x - ∂M/∂y)<k>; if curl <F> = 0 then <F> is irrotational |
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Test for Conservative Vector Field In Space |
<F> is conservative if and only if curl <F> is zero |
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Divergence of a Vector Field |
∇⋅<F(x,y,z)> = ∂M/∂x + ∂N/∂y + ∂P/∂z; if div <F> = 0 then <F> is solenoidal |
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Evaluation of a Line Integral as a Definite Integral |
∫ƒ(x,y,z)ds = ∫ƒ(x(t),y(t),z(t))√([x'(t)]²+[y'(t)]²+[z'(t)]²)dt, note ds = ||<r'(t)>||dt |
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Line Integral of Vector Field |
∫<F>⋅<dr> =∫<F>⋅<T>ds = ∫<F((x(t),y(t),z(t))⋅<r'(t)>dt |
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Fundamental Theorem of Line Integrals |
∫<F>⋅<dr> = ∫∇ƒ⋅<dr> = ƒ(x(b),y(b)) - ƒ(x(a),y(a)) |
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Independence of Path and Conservative Vector Fields |
∫<F>⋅<dr> is independent of path if and only if <F> is conservative |
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Equivalent Conditions of Conservative Vector Fields |
1. <F> is conservative. That is <F> = ∇ƒ for some ƒ. 2. ∫<F>⋅<dr> is independent of path. 3. ∫<F>⋅<dr> = 0 0for every closed curve in R. |
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Green's Theorem |
∫(Mdx + Ndy) = ∫∫(∂N/∂x - ∂M/∂y)dA for simply connected regions with smooth boundaries |
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Line Integral Area |
If R is a place region bounded by a piecewise smooth simple closed curve C,, oriented counterclockwise, then the area of R is given by A = ½∫(xdy - ydx) |
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Green's Theorem First alternate form |
∫(Mdx + Ndy) = ∫∫(curl <F>) ⋅ <k>dA for simply connected regions with smooth boundaries |
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Green's Theorem Second alternate form |
∫(Mdx + Ndy) = ∫∫(div <F>)dA for simply connected regions with smooth boundaries |
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Parametric Surface |
<r(u,v)> = x(u,v)<i> + y(u,v)<j> + z(u,v)<k> |
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Normal Vector to a Smooth Parametric Surface |
<N> = (∂<r>/∂u) x (∂<r>/∂v) = |[<i>, <j>, <k>], [∂x/∂u, ∂y/∂u, ∂z/∂u], [∂x/∂v, ∂y/∂v, ∂z/∂v]| |
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Area of a Parametric Surface |
Surface area = ∫∫dS = ∫∫||<N>||dA |
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Evaluating a Surface Integral |
∫∫ƒ(x,y,z)dS = ∫∫ƒ(x,y,g(x,y))√(1 + [∂g(x,y)/∂x]² + [∂g(x,y)/∂y]²)dA |
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Flux Integral |
∫∫<F>⋅<N>dS |
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Flux Integral for an Upward Surface |
∫∫<F>⋅<N>dS = ∫∫<F>⋅(-∂g(x,y)/∂x<i> - ∂g(x,y)/∂y<j> + <k>)dA |
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Flux Integral for an Downward Surface |
∫∫<F>⋅<N>dS= ∫∫<F>⋅(∂g(x,y)/∂x<i> + ∂g(x,y)/∂y<j> - <k>)dA |
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The Divergence Theorem |
∫∫<F>⋅<N>dS = ∫∫∫div<F> dv |
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Souce Field |
div <F> > 0 |
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Sink Field |
div <F> < 0 |
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Incompressible Field |
div <F> = 0 |
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Stokes's Theorem |
∫<F>⋅<dr> = ∫∫(curl <F>)⋅<N> dS |
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