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Line Integral
If r(a,b) is a smooth parametreisation of a path C and F:Omega -->r^n is a vector field such that F is defined and bounded on the image of r than we define the integral of F.dr over the path C as the integral between a and b of F(r(t)).r'(t)dt
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Fundamental Theorem of Algebra for scalar fields
Let Omega be a subset of R^n and f:Omega ->R^n be a smooth scalar feidl and assume that r(a,b)-->R^n is a piecewise smooth parameterisation of a path C whose image is included in Omega. Then the integral of nablaf.dr over C is equal to f(r(b))-f(r(a)
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Potential
If F=nabla f and f is smooth then the line integral of F.dr over C does not depend on the path C but just on the end points. f is the potential of F.
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Conservative
F:O->R^n is called conservative if its line integral does not depend on path. If for any C1, C2 piecewise smooth paths in Omega with r,s:(a,b)->O with r(a)=s(a) and r(b)=s(b) we have that the line integrals of F.dr over C1 and F.ds over C2 are equal
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Conservative VFs
If F is a vector field that is continuous on a connected set Omega subset of R^n then the following are quivalent: 1. There exists a scalar field such that F=nablaf, 2. F is conservative 3. The integral along a closed path is 0
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Example of an integral that = 0 but F is not conservative
F(x,y)=(xy,y) r(t)=(acost,asint) t e [0,2pi]
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Curl for conservative VFs
Recall that curl(nablaf)=0 and so if F=nablaf then curlF = 0
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Property of conservative VFs
Let F:O->R^n be a conservative VF whose components are smooth. For all i,j=1,..,n dFi/dxj=dFj/dxi (dF1/dy=dF2/dx)
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Example of a vector field that has the property that dF1/dy=dF2/dx but it is not conservative
F(x,y)=(-y/x^2+y^2,x/x^2+y^2) r(t)=(cost,sint)
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Star shaped domain theory
If O subset of R^n is a star-shaped set, F is a smooth VF with the property from before then F is conservative. The potential is defined as f(x)=integral between 0 and 1 of F(1-t)x0+tx).(x-xo)dt
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Greene's Theorem
Let C be a piecewise smooth simple closed curve (SSCC) in the plane with param r[a,b]->R^2 that traveses C anti-clockwise and F is a smooth vector field defined in D vounded by C. Then work done = double integral over D dQ/dx-dP/dy dxdy
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Green Corollary 1
If omega in R^2 is s.t it is bounded by a simple plane curve C and F(P,Q):O->R^2 is a VF s.t dQ/dx=dP/dy then F is conservative
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Green Corollary 2
If C is a simple planar closed curve and omega is the region bounded by C then the area of omega is cSxdy=-cSydx=1/2cS-ydx+xdy
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Area of a surface
double integral over omega of |r_a x r_b| dadB
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Surface integrals of first kind
Let O C R^2 and r:O->S be a smooth param of surface S and f:S>R a smooth scalar field. The integral of f on S is sSSfds= oSSf(r(a,b))|ra x rb| dadb
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Surface integral of second kind
Let r:O>R^3 be a smooth param of an orientable suface S and F:S>R^3 a smooth VF. The intgral of F on S is sSSF.nds=oSSF(r).(ra x rb) dadB
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Stokes preamble 1
Suppose D subset of R^2 is a region in the plane that is bounded by a simple piecewise smooth curve T subset D. Suppose r:D>R^3 is apiecewise smooth parameterisation of orientable S and C=r(T).
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Stokes preamble 2
Suppose als o that C is oriented so that it inherits the counterclockwise orientation of T through r. Finally let F:S>R^3 be asmooth VF
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Stokes Formula
Integral of F.dr over a closed curve C = double integral of curlF.ndS over S
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Gauss
Let V be a solid in R^3 that is bounded by a piecewise smooth orientable S and n is the normal (outward). If F:V>R^3 is a smooth VF then sSSF.nds=vSSSdivFdxdydz
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Card 2

Front

Let Omega be a subset of R^n and f:Omega ->R^n be a smooth scalar feidl and assume that r(a,b)-->R^n is a piecewise smooth parameterisation of a path C whose image is included in Omega. Then the integral of nablaf.dr over C is equal to f(r(b))-f(r(a)

Back

Fundamental Theorem of Algebra for scalar fields

Card 3

Front

If F=nabla f and f is smooth then the line integral of F.dr over C does not depend on the path C but just on the end points. f is the potential of F.

Back

Preview of the back of card 3

Card 4

Front

F:O->R^n is called conservative if its line integral does not depend on path. If for any C1, C2 piecewise smooth paths in Omega with r,s:(a,b)->O with r(a)=s(a) and r(b)=s(b) we have that the line integrals of F.dr over C1 and F.ds over C2 are equal

Back

Preview of the back of card 4

Card 5

Front

If F is a vector field that is continuous on a connected set Omega subset of R^n then the following are quivalent: 1. There exists a scalar field such that F=nablaf, 2. F is conservative 3. The integral along a closed path is 0

Back

Preview of the back of card 5
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