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15 Cards in this Set
- Front
- Back
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What is the vector equation of a plane? |
r•n=a•n r and a are distinct points on the plane n is the normal to the plane |
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Give the equation of the directional derivative of f(x,y,z) at the point (a,b,c) in the direction u as a limit as t->0 |
Duf(a,b,c) = limt->0 [f(a+tu1 ,b+tu2 , c+tu3) - f(a,b,c)]/t |
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Give the directional derivative of f(x,y,z) at the point (a,b,c) in the direction u in terms of grad(f) |
u•grad(f) |
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What are the conditions under which the double integral of [c×f(x,y)+d×g(x,y) dxdy] can be split into two separate single integrals |
f and g are continuous and c and d are constant |
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Define the curl of a vector field F = Mî +Nj^ |
Curl(F) = Nx - My =∇x⃗F⃗⃗ |
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Give the divergence of a vector field F = Mî + Nj^ |
div( F ) = Mx + Ny =∇·⃗F⃗ |
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Give the divergence of a vector field F = Mî + Nj^ +Rk^ |
div( F ) = Mx + Ny + Rz = ∇·⃗F⃗ |
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Suppose S is a closed surface oriented outwards that encloses a solid domain V. If F is a differntiable vector field in V then |
∫∫ F · d S = ∫∫∫ᵥ div( F ) dV |
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Give the curl ( ⃗F⃗ ) for a vector field ⃗F⃗ = Nî + Mj^ + Rk^ as a cross product |
curl ( ⃗F⃗ ) = ∇ x ⃗F⃗ |
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If F is conservative on a simply connected domain D ⊂ R³ what can be said about the curl of F |
Its zero |
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State the fundamental theorem for line intergrals |
Let C be a smooth curve given by the position vector ⃗r⃗ (t), a≤t≤b. Let f be differentiable function whose gradient ∇f is continuous on C. Then ∫ c ∇f·d⃗r⃗ = f(⃗r⃗(b)) - f(⃗r⃗ (a)) |
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state Greens theorem in tangent form |
Let C be a closed curve oriented clockwise and enclosing a region R ⊂(reals). If ⃗F⃗ is a vector field defined and differentiate everywhere on R, then ∫ c ⃗F⃗ ·d⃗r⃗ = ∫∫ R curl(⃗F⃗ ) dA |
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state Greens theorem in normal form |
Let C be a closed curve oriented clockwise and enclosing a region R ⊂(reals). If ⃗F⃗ is a vector field defined and differentiate everywhere on R, then ∫ c ⃗F⃗ ·ňds = ∫∫ R div(⃗F⃗ ) dA |
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State gauss divergence theorem |
Let D be a closed surface oriented outwards and enclosing a solid region D R³. If F be a vector field defined and differentiable in D, then ∫∫ s F·d⃗S⃗ = ∫∫∫ D div(⃗F⃗)dV |
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State Stokes' theorem |
Let C⊂R³ br a closed curve and let S be any surfave bounded by C. Assume the orientations of C and S are compatible. If ⃗F⃗⃗ be a vector field defined and differentiable in S, then ∫c F⃗ ·d⃗r⃗ = ∫∫s (∇x⃗F⃗ )·n^dS |
