# KCL Calculus 2 class test 3 formula

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Integral of scalar function over curve
∫f(r(t))|r’(t)|dt
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Arc length
∫|r’(t)|dt
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Line integral (vector fields)
∮v.dr=∫v(r(t)).r’(t) dt
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Integral of ∇f
∫∇f=f(q)-f(p) where C∈⟦p,q⟧
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what does v=∇f mean?
∮v.dr=0 (closed arc)
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Area of U?
∫∫1.dx.dy
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Integral when changing variables?
∫∫ ||J||du.dv
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Polar co-ordinates change of variables?
∫∫f(rcosθ,rsinθ)r.dr.dθ
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What is Green's theorem?
∮v.dr=∫∫((dQ/dx)-(dP/dy))dxdy
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What are P and Q in Green's theorem?
V(x,y)=P(x,y)i+Q(x,y)j
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What is the test for a gradient field?
∮v.dr=0
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What is the alternative formula for area (x,y)?
1/2∮(-y.dx+x.dy)
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What is the parameterisation of a graph surface (x,y)?
xi+yj+f(x,y)k
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What is the formula for the fundamental vector product (u,v)?
N(u,v)=dr/du x dr/dv (cross product)
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Fundamental vector product (x,y)?
-df/dx(i)-df/dy(j)+k for r=(x,y,f(x,y))
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What is the significance of N(u,v)?
Perpendicular to surface at r(u,v)
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What's the formula for a surface integral (u,v)?
∫∫g(r(u,v))||N(u,v)||dudv
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What's the formula for a surface integral (x,y)?
∫∫f(x,y)√(1+(df/dx)^2+(df/dy)^2)dxdy for z=f(x,y)
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Area of surface (u,v)?
S=∫∫|N(u,v)|
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Area of surface (x,y)?
∫∫√((df/dx)^2+(df/dy)^2+1)dxdy
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Formula for flux?
∫∫v.ndσ=∫∫v.N(u,v) where v=vector field
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Formula for flux (z=f(x,y))?
∫∫(-v1(df/dx)-v2(df/dy)+v3)dxdy
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What is v in the flux formula?
v=v1i+v2j+v3k across S in direction of upwards normal
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What is Stoke's theorem?
∫∫(∇xv).ndσ=∮v.dr
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What does it mean when surface is closed (Stoke's theorem)?
∫∫(∇xv).ndσ=0
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What does cos^2(t) equal?
1/2(1+cos2t)
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what does sin^2 equal?
1/2(1-cos2t)
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## Other cards in this set

Arc length

∫|r’(t)|dt

### Card 3

#### Front

Line integral (vector fields)

Integral of ∇f

### Card 5

#### Front

what does v=∇f mean?