Calc III

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jesperez10
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246322
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Calc III
Updated:
2013-11-18 22:04:07
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Calculus III
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Chapter 12: Vectors and the Geometry of Space (and more)
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  1. 14.4
    Explain the process to find an eq. for the tangent plane.
    Given: f(x,y), pt.
    • 1. Find the gradient vector.
    • 2. Plug the point into the
    • 3. Insert given pt. and gradient vector into the equation of a plane (where z=-1).
  2. 14.6
    Find the directional derivative.
    Given: f(x,y), vector, and pt.
    • 1. Find the gradient vector 
    • 2. Find the unit vector of the given pt.
    • 3. dot product of 1. and 2.
  3. 14.7
    In which direction is the rate of change of f(x,y) maximized.
    Given: gradient vector.
    unit vector of the gradient vector
  4. 14.7
    Find the max rate of change.
    Given: gradient vector
    magnitude of the gradient vector.
  5. 14.8
    Find the max value of f(x,y) subjected to g(x,y).
    • 1. Find the gradient vector with respect to f(x,y) and g(x,y).
    • 2. Solve of λ. 
    • 3. Solve for x or y.
    • 4. Substitute to find y or x.
    • 5. Find the max by substituting x and y into g(x,y).
  6. 12.5
    Find and eq. for the plane that passes through a pt. and contains a parametric equation.
    • 1. Dot product of Q and R.
    • 2. QP X V.
    • 3. answer in eq. form.
  7. 14.4
    Find the linearization of f(x,y) at pt.
    • 1. Find the gradient vector.
    • 2. Find z using pts in f(x,y).
    • 3. answer in eq. form
  8. 15.5
    What is the mass m using the applications of double integrals?
    p(x,y) dA; where the domain is D
  9. 15.5
    What is the moment of the entire lamina about the x-axis using the applications of double integrals?
    Mx=y * p(x,y) dA; where the domain is D
  10. 15.5
    What is the moment of the entire lamina about the y-axis using the applications of double integrals?
    My=x * p(x,y) dA; where the domain is D
  11. 15.5
    What is the polar moment of inertia using the applications of double integrals?
    Io(x2+y2) p(x,y) dA; where the domain is D
  12. 15.5
    What is the moment of inertia of the lamina about the x-axis using the application of double integrals?
    Ix(y2)p(x,y) dA; where the domain is D
  13. 15.5
    What is the moment of inertia of the lamina about the y-axis using the application of double integrals?
    Iy=(x2) p(x,y) dA; where the domain is D
  14. 15.9
    What is the formula for triple integration in spherical coordinates?
    • E= f(psinΦcosθ, psinΦsinθ, pcosΦ)p2sinΦ dpdθdΦ;
    • where E{(p,θ,Φ)|a≤p≤b, α≤θ≤β, c≤Φ≤d}
  15. 15.9
    Explain the triple integrals with spiracle coordinates using a diagram:
  16. 15.9
    In spherical coordinates, what is x? y? z?
    • x= psinΦcosθ
    • y= psinΦsinθ
    • z=pcosΦ
  17. What is C?
    • A smooth space curve given by the parametric equation:
    • x=x(t)  y=y(t)  z=z(t)

    or my a vecor:

    r(t)=x(t)+y(t)j +z(t)k
  18. 16.2
    Evaluate a line integral, where C is the given curve.
    Given: f(x,y,z),x,y,z, domain of t.
    • 1. f(x(t),y(t),z(t))√(dx/dt)2 +(dy/dt)2 +(dz/dt)2)dt
  19. 16.2
    What is the representative of the line segment that starts at r0 and ends at r1?

    r(t)=(1-t)r+ tr1; where rand rare point on the line.
  20. 16.2
    Evaluate the line integral  F dr, where C is given by the vector functions r(t).
    Given: F(x,y), r(t)
    • 1. Remember that r(t) goes into F.
    • *F is the normal vector of the curve
    • 2. Find r'(t)
    • 3. Plug x and y of r into eq. F(r(x,y)) 
    • 4. Combine x's with x's and y's with y's
    • 5. use W=  F(r(t)) ⋅ r'(t)dt
  21. 16.3
    How do you determine if the vector field is a conservative function?
    δP/δy =δQ/δx; throughout D
  22. 16.3
    If F(x,y)= some i+j, what are the steps to find a function such that F=
    • 1. Find δP/δy =δQ/δx to be conservative.
    • 2. Find the partial with respect to x and y. i.e.<i,j>
    • 3. with respect to x.
    • 4.  Take the partial with respect to y of 3..
    • 5.  g'(y) with respect to y. 
    • 6 .Combine 3. and 5..
  23. 16.3
    What is the fundamental theorem of line integrals?
     F⋅dr = f(end)-f(start)
  24. 16.4
    What is the Green's Theorem?
     Pdx+Qdy=
  25. ?
    "del" =
  26. 16.5
    What is the curl?
     ;where = <i,j,k> and F= P,Q,R
  27. 16.5
    curl(f)=
    0
  28. 16.5
    div F=

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