MTH2010 Partial Derivatives

MTH2010 Partial Derivatives

1. Review of 1030

   • Dot product
     • Scalar that defines angle b/w a and b
       • Orthogonal > a.b=0
       • cos theta= a.b / mag(a)mag(b)
       • Resolutes
         • Vector of a in direction b
           • b=a.b hat b hat
         • Scalar of a in direction b
           • b=a. b hat
         • Perpendicular Resolute
           • a- vector resolute
   • Cross Product
     • Box Product
       • mod ((axb).c)
       • Measures volume
     • Produces vector c, perp to a and b
     • Right handed
     • mod(a x b)=area of parallelogram spanned by a and b
     • mag(a) mag(b) sin theta= mag( a x b)
     • bxa=-axb
     • for i j k triangle clockwise is positive
     • For crossing two vectors of three variables use det
       • Break into 3 dets with i j and k
         • add the i and k term and minus the j
   • Lines
     • r(t)=v+td
     • Shortest distance at line.point=0
     • Symettric equations
       • (x-x0)/a =(y1-y0)/b=(z-z0)/c
   • Planes
     • n.(r-r0)=0
     • z=z0-a/c u-b/c v
       • Parametrically r=<x0+u,yo+v,z0-a/c u -b/c v

2. R3

   • A point requires three real scalars as defined by an order triplet
   • Divide plane into octants

3. Surfaces

   • Sketching Surfaces
     • Traces
       • Contour Plot/Level curves
         • Traces parralel to the xy plane
           • Let z equal a constant , k
       • Intersections of the surface with planes parallel to the coordinate axis
         • i.e letting x=0 or y=0 or z=0
     • Plot intercepts
       • Spheres, ellipsoids and planes
   • Types of surfaces
     • Sphere
       • (x-h)^2+(y-k)^2+(z-k)^2=r^2
     • Cylinder
       • A curve in a plane which is extended normal to the plane
         • x^2+y^2=1
     • Standard forms
       • ax^2+by^2+cz^2+d=0
         • inc. ellipsoids,hyperboloids and cones
           • Ellipsoid : x^2+y^2+z^2=1
             • If=k is a hemisphere
           • Cone
             • x^2+y^2=z^2 which opens along the lone variable
           • Hyperboloid
             • 1 sheet
               • x^2+y^2-z^2=1
                 • The negative variable gives the axis centered on
             • 2 sheet
               • -x^2-y^2+z^2=1
                 • The positive variable gives the axis centered on
                 • Disjointed
       • ax^2+by^2+cz=0
         • elliptic paraboloids and hyperbolic paraboloids
           • Elliptic paraboloids
             • z=x^2+y^2
               • Opens on non squared variable
           • Hyperbolic Paraboloid (Saddle)
             • z=x^2-y^2
   • Parametrisation
     • If possible, you can integrate that surface

4. Limits and Continuity

   • f(x,y) has limit L if for every eplison>0 there exists row>0 such that mod(f(x,y)-L)< epsilon wherever mod(sqrt( (x-a)^2 +(y-b)^2),<row
     • To show limit does not exist find inconsistancy
   • Continuity
     • f(a,b)= lim as (x,y) tends to (a,b) of f(x,y) must exist for continuity

5. Partial Derivatives

   • fx (x,y)= lim as h tends to 0 of1/h .f(x+h,y)-f(x,y)
   • fy (x,y)= lim as k tends to 0 of1/h .f(x,k+y)-f(x,y)
   • fy and fx do not need to be the same
   • fx is derivative of level curve at x=a. fxx is the curvature of level curve at x=a
   • fy is the derivative of level curve at y=b and fyy is the curvature of level curve at y=b
   • fyx is the rate of change of fy(x,b) at x=a
     • fyx is rate of change of fx(a,y) at y=b
   • Clairauts Theorem
     • If f is defined on disk D which contains (a,b) and fx and fy are cont. on D then : fxy =fyx
   • PDE's
     • Laplace
     • Wave
     • Heat

6. Linear Approximations

   • Differential
     • dz=dz/dx dx+ dz/dy dy
       • f(x+dx,y+dy) is approximated to f(a,b)+dz
   • Approximation
     • Linear Approximation
       • L(x,y)=f(a,b)+fx(a,b)(x-a)+fy(a,b)(y-b)
         • If fx and fy exist near (a,b) and are cont. at (a,b)
     • Quadratic Approximation
       • Better than linear
         • In conjunction with Taylor expansion
           • First order (see notes)
           • 2nd order (see notes)

7. Chain rule (see notes)

8. Implicit Differentiation

   • dy/dx= -dF/dx / dF/dy

9. Directional Derivatives

   • Duf= grad f . u hat
     • u is direction vector
     • Directional derivative of f in direction u (scalar)
       • Max occurs when u hat = grad f hat ; i.e pointed in the direction of the gradient
     • Gradient vector
       • Points in direction of maximum rate of change,where the maximum rate of change is duf
       • Is the vector with components fx and fy
       • Normal to contours
       • Slope of the directional derivative
     • Max rate of change occurs when mag Duf =mag grad f

10. Tangent Planes

    • Grad f defines normal to the surface
    • See notes for general form

11. Maximum and Minimum

    • Lagrange Multipliers
      • solving a problem where a max/min of a funtion is constrained by another function
        • g(x,y)=c
          • At min/max along constraint, the tangent to g is perp to grad f
        • lamda grad g=grad f
        • Two constraints g and h
          • grad f =lamda grad g + mui grad h
    • Local at fy=fx=0
      • This collapses tangent plane to a horizontal plane so z=f(a,b)
      • D=fxx.fyy-(fxy) squared
        • D>0
          • fxx>0
            • Local min
          • fxx<0
            • Local max
        • D<0
          • Saddle
        • D=0
          • Test Fails
        • Global max/min on bounded dom
          • Find values at critical points,find extremes of f on boundary,largest values of these is a maximum and smallest is a minimum