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projuv
Write equation and projection of _ onto _ |
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[image]
part of v in direction of u |
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How do you check if points are coplanar? |
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Volume of parallepipid is 0 |
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Find an intersection between 2 lines and if they are skew or not
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1) Parametrize w/ respect to t and s
2) Solve system of equations with t and s
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Find the equation of a plane containing a line |
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use the direction vector of the line and then pick a point on
line to point given |
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Line of intersection of 2 planes |
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1)Cross the normal vectors
2) Find a point on both planes to use in eqn |
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Write where a function is continuous using good notation
ex y>0 |
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fxy dydx or dxdy
fyx dydx or dxdy
Clairauts theorem states that___
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fxy dydx
fyx dxdy
The two things above are equal |
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| Directional Derivatives in vector (a,b) |
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[image]
(a,b) MUST be a unit vector |
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| See if function is a max/min/saddle |
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[image]
D > 0
fxx<0 -> max
fxx>0 -> min
D < 0
saddle pt |
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| Find the arclength of r(t) |
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r(t) = 〈x(t),y(t)〉,t∈[a,b]
[image] |
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| Given p(x,y) as a density, what's the mass of a thin wire? |
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Given [image] and [image]
another way of writing [image]
would be? |
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| [image] which equals [image] |
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| Fundamental Thrm for Line Integrals |
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Can draw a disc around every point that lies completely in P.
[image] |
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| Connected vs Simply Connected |
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Connected -> Given 2 points in region D, there can be a path drawn between them that stays in D
Simply Connected -> Same as above but NO holes allowed |
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| Conditions for Cons. Vector Field |
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| Qx=Py AND Open simply connected region D |
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| Mass and CM using double integrals given density p |
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| Volume under a graph (2 choices) |
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| Area of a region D defined by x and y |
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| Avg Value of a function f(x,y) |
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| Spherical Coordinates Order (p,a,b)? |
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| Surface integral definition |
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| Flux through a surface given F vector field |
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| [image]\iint_S \vec{F} \cdot dS |
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| How do you change the variables in a parallelogram? |
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1) x = au+bv and y = cu + dv
2) make (or be given) lines in the form of n = px+ qy (n p and q are constants)
3) Plug in au+bv and cu+dv into line eqn, solve for abcd |
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| State Green's Theorem and the conditions necessary |
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CCW Simple Closed Curve
[image] |
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| Greens thrm - how do you deal with holes |
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| Have outside oriented CCW and inside oriented CW. ADD!! the two together |
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What is the curl of [image]
curl of the gradient? |
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| Given z=g(x,y) and F = , what would be the better way to calculate [image] |
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| Stokes Thrm - State Conditions |
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| Simple Closed Curve [image] |
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| Normal Vector to a parametrized surface is |
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Orientation -> How to do it for stokes?
ex: bottom of a hemisphere |
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| Use normal vector and RHR. Bottom of a hemisphere means CW, top means CCW |
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| Divergence theorem -> state condition and theorem |
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simple solid with outward orientiation for normal
[image] |
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rate of circulation per unit area
infinitely small rotation of vector field |
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| Net rate of vector field expansion |
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div < 0
Vector lines that END at the point are LONGER than the ones that START at the point.
----> P -> |
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div > 0
Vector lines that END at the point are SHORTER than the ones that START at the point.
-> P -----> |
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| Reparametrize curve with respect to arclength |
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Solve for s.
[image]
the MAGNITUDE of r'u = sqrt(components^2) |
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