Intermediate Calc II Definitions

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Intermediate Calc II Definitions
2015-04-21 21:23:26

Intermediate Calculus Definitions
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  1. Vector Addition
    If u and v are vectors position so that the initial point of v is at the terminal point of u, then the sum u+v is the vector from the initial point of u to the terminal point of v
  2. Scalar Multiplication
    If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose magnitude is |c| times the magnitude of v and whose direction is the same as v if c>0 and is opposite to v if c<0. If c=o or v=0, then cv=0.
  3. Two nonzero vectors are parallel if
    they are scalar multiples of one another
  4. Normal Vector
    Any vector that is perpendicular to the plane. The normal vectors of a plane containing vectors u and v are all the scalar multiples of uxv
  5. For all ε>0, there exists δ>0 such that (x,y)∈Domainf and ⇒|f(x,y)-L|<ε
  6. A function f of two variables is continues at (a,b) if
    • For D⊆R2, we say f is continuous on D if f is continuous at every point (a,b)∈D
  7. For all ε>0, there exists δ>0 such that if (x,y,z)∈Domainf and  ⇒|f(x,y,z)-L|<ε
  8. For functions of n variables, define 
    Let x=<x1,x2,...,xn> and a=<a1,a2,...,an> be vectors in Rn. If f is defined on a subset D of Rn, then  means that for all ε>0, there exists δ>0 such that if x∈D and 0<x-a<δ⇒f(x)-L<ε
  9. If z=f(x,y), then f is differentiable at (a,b) if
    • Δz can be expressed in the form:
    • Δz=fx(a,b)(Δx)+fy(a,b)(Δy)+ε1Δx+ε2Δy
    • where ε1→0 and ε2→0 as (Δx,Δy)→(0,0)
  10. Limit Definition of the Directional Derivative
    • The Directional Derivative of f(x0,y0) in the direction of a unit vector u=<a,b> is
    • , provided the limit exists
  11. Gradient Vector
    If f(x,y), ∇f(x,y)=<fx(x,y), fy(x,y)>=
  12. Local Maximum and Absolute Maximum
    • f(a,b) is a local maximum value if f(x,y)≤f(a,b) for all points (x,y) in some disk with center (a,b)
    • If this inequality holds for all points (x,y) in the domain of f then f has an absolute maximum at (a,b)
  13. Local Minimum and Absolute Minimum
    • f(a,b) is a local minimum if f(x,y)≥f(a,b) when (x,y) is near (a,b).
    • If this inequality holds for all points (x,y) in the domain of f then f has an absolute minimum at (a,b)
  14. Critical or Stationary Point
    a point (a,b) is a critical point of f if fx(a,b)=0 and fy(a,b)=0, or if at least one of these partial derivatives does not exist.
  15. Normal Line to a surface S in R3
    The line through P which is perpendicular to the tangent plane to S at P. The direction of the normal line is given by ∇F<x0,y0,z0>
  16. Double Integral of f over the rectangle R is
    , if the limit exists
  17. Triple integral of f over the box B
    if the limit exists
  18. Jacobian of the Transformation T given by x=g(u,v) and y=h(u,v) is