Math 241: Multivariable calculus
                               Professor Leininger
         www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
                                    Fall 2014
        www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
                           Calculus of 1 variable
       In Calculus I and II you study real valued functions
                                    y = f(x)
       of a single real variable.
       Examples:
         • f (x) = x2, r(x) =    2x2+x  , h(θ) = sin(θ) + cos(2θ),
                               x3−5x+20
            g(u) = eu,...
         • T(t) = temperature in Champaign-Urbana, t hours after
            midnight on August 25.
         • ρ(d) = density of a piece of wire at distance d from one end.
        www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
               Three key concepts from Calculus I, II.
       f (x), a function of one variable.
        1. The derivative: f ′(x) = df = d f (x) = dy.
                                    dx    dx        dx
              • Rate of change.
              • Slope of the tangent line to the graph.
        2. The integral: Rb f (x)dx.
                           a
              • Signed area under graph.
              • Average value  1  Rbf(x)dx.
                              b−a a
        3. Fundamental Theorem of Calculus: Relates the two.
              • f(b)−f(a) = Rbf′(x)dx.
                               a
        www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
                      1 variable is too constrained
       Functions of a single variable are insufficient for modeling more
       complicated situations.
       Examples:
         • The temperature depends on location as well as time. Need
            to specify location, e.g. by latitude x and longitude y, and
            time, e.g. t hours after midnight:
               T(x,y,t) = temperature at time t in location (x,y).
         • Density of a flat sheet of metal can depends on the point in
            the sheet, specified by x and y coordinates
               δ(x,y) = density of point at (x,y) in a sheet of metal
        www.math.uiuc.edu/∼clein/classes/2014/fall/241.html