Calc III - Ch13 formulae.txt

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Author:
broach13
ID:
235892
Filename:
Calc III - Ch13 formulae.txt
Updated:
2013-09-22 17:03:12
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calc iii formula
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formula for calc iii ch 13
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  1. Arc length of a vector function <x(t), y(t), z(t)>
  2. Unit tangent vector T of r(t)
  3. Curvature given arc length parametrization and T
  4. How to relate arc length to r(t)
    ds/dt = ||r'(t)||
  5. Curvature given T and r(t)
  6. Curvature given only r(t)
  7. Curvature of a plane curve given y = f(x)
  8. Find given only r(t)
  9. Find B given N and T
    B = N x T
  10. How to find the normal plane
    Use T(t) as the normal and r(t) for a given point t on the curve
  11. How to find the osculating plane
    Use B(t) as the normal and r(t) for a given point t
  12. v(t) given r(t)
    v(t) = r'(t)
  13. a(t) given r(t)
    a(t) = r''(t) = v'(t)
  14. v given T
    V = vT = ||v||T
  15. Tangential component of acceleration given r(t) and T
    ||r'(t)||T = T
  16. normal component of acceleration given r(t)
  17. Distance formula in R3
  18. Equation of a sphere in R3 with radius r
  19. How to compute the projection of a figure onto a given plane
    Drop the non-included coordinate to 0
  20. A dot A
    ||A||^2
  21. a dot (b+c)
    a dot b + a dot c
  22. c(a dot b)
    c(a) dot b = a dot cb
  23. Angle between vectors
  24. Two vectors are orthogonal iff
    a dot b = 0
  25. How to find direction cosines
    ||a||<cos alpha, cos beta, cos gamma>
  26. scalar projection of b onto a
  27. vector projection of b onto a
     (scalar proj * unit vector of a)
  28. identity for the angles of direction cosines
  29. a x a
    0-vector
  30. Relating a x b to the angle
    a x b = ||a||||b||sin(theta)
  31. Two nonzero vectors are parallel iff
    a x b = 0
  32. Area of a parallelogram determined by a and b
    A = a x b
  33. T/F the cross product is commutative
    F
  34. (ca) x b =
    c(a x b) = a x (cb)
  35. a x (b+c)
    a x b + a x c
  36. (a + b) x c =
    a x c + a x b
  37. a dot (b x c) =
    (a x b) dot c
  38. a x (b x c) =
    (a dot c)b - (a dot b)c
  39. Volume of a parallelepiped
    Scalar triple product = |a dot (b x c)| (l, w, h)
  40. Torque =
    r x F = ||r|| ||F|| sin(theta)
  41. Area of a triangle with vectors at a common vertex
    area = |a x b|
  42. Vector-parametric equation of a line
    r(t) = r(0) + vt
  43. Symmetric eqn of a line
    • (x - x0)/a = (y - y0)/b = (z - z0)/c
    • If a, b, or c=0, then the equality between the other two is preserved and we get something like x = x0, y(terms) = z(terms)
  44. Vector--parametric equation of a line segment
    r(t) = (1-t)r0 + tr1
  45. How to verify skew lines
    First, ensure that they never intersect (in space, not necessarily in time). Then ensure that they aren't parallel
  46. Vector equation of a plane
    n dot r = n dot r0
  47. scalar eqn of plane
    • a(x-x0) + b(y-y0) + c(z-z0) = 0
    • n = <a,b,c>
  48. linear eqn of a plane
    ax + by + cz + d = 0
  49. How to determine parallelity of planes
    if their normal vectors are parallel
  50. How to determine angle between 2 planes
    use the normal vectors and the angle-dot-product formula
  51. How to find the line of intersection of 2 planes
    Solve for one of the variables (x,y,z) in terms of the others and use it as a parameter
  52. Distance from a point to a plane
    (or between 2 parallel planes)
  53. Distance between skew lines
    They define 2 planes. You just need a vector orthogonal to both skew lines, so cross them! That's n

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