� � MIT 3.016 Fall 2005 � W.C Carter Lecture 12 c70 Oct. 14 2005: Lecture 12: Multivariable Calculus Reading: Kreyszig Sections: 8.5 (pp:428–33) , 8.8 (pp:444–446) §8.9 (pp:446–452) § §The Calculus of Curves In the last lecture, the derivatives of a vector that varied continuously with a parameter, �r(t), were considered. It is natural to think of �r(t) as a curve in whatever space the vector �r is defined. In this way, a curve is represented by N coordinates as a single value takes on a range of numbers. Mathematica r� Example: Lecture-12 Curves, Tangents, Surfaces Visualizing a curve Because the derivative of a curve with respect to its parameter is a tangent vector, the unit tangent can be defined immediately: d�r ˆdt (12-1)u = d�r dt � It is convenient to find a new parameter, s(t), that would make the denominator in Eq. 12-1 equal to one. This parameter, s(t), is the arc-length: t s(t) = ds to t = ��dx2 + dy2 + dz2 to (12-2)t�� dx dy dz = ( )2 + ( )2 + ( )2dt dt dt dtto t �� d�r d�r = ( dt ) · ( dt )dt toMIT 3.016 Fall 2005 � W.C Carter Lecture 12 c71 and with s instead of t, d�r ˆu(s) = (12-3)ds This is natural because ��r� and s have the same units (i.e., meters and meters, foots and feet, etc) instead of, for instance, time, t, that makes d�r/dt a velocity and involving two different kinds of units (e.g., furlongs and hours). With the arc-length s, the magnitude of the curvature is particularly simple, dˆu d2�r κ(s) = � (12-4)ds � = � ds2 � as is its interpretation—the curvature is a measure of how rapidly the unit tangent is changing direction. Furthermore, the rate at which the unit tangent changes direction is a vector that must be u u = 1) = 0) and therefore the unit normal is defined by: normal to the tangent (because d(ˆ ˆ· 1 dˆu pˆ(s) = (12-5)κ(s) ds There two unit vectors that are lo cally normal to the unit tangent vector ˆ(s) and the curve u�p(s) × ˆ u(s) × pˆ. This last choice is called the unit binormal, ˆb ≡ ˆunit normal ˆ u and ˆ u(s) × pˆand the three vectors together form a nice little moving orthogonal axis pinned to the curve. Furthermore, there are convenient relations between the vectors and differential geometric quantities called the Frenet equations. qcqckUsing Arc-Length as a Curve’s Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . However, it should be pointed out that—although re-parameterizing a curve in terms of its arc-length makes for simple analysis of a curve—achieving this re-parameterization is not necessarily simple. Consider the steps required to represent a curve �r (t) in terms of its arc-length: integration The integral in Eq. 12-2 may or may not have a closed form for s(t). If it does then we can perform the following operation: inversion s(t) is typically a complicated function that is not easy to invert, i.e., solve for t in terms of s to get a t(s) that can be substituted into �r(t(s)) = �r(s). These difficulties usually result in treating the problem symbolically and the resorting to numerical methods of achieving the integration and inversion steps.MIT 3.016 Fall 2005 � W.C Carter Lecture 12 c72 Mathematica r� Example: Lecture-12 Calculating arclength Finding arc-length s of a curve and using as parameter Scalar Functions with Vector Argument In materials science and engineering, the concept of a spatially varying function arises frequently: For example: Concentration ci(x, y, z) = ci(�x) is the number (or moles) of chemical species of type i per unit volume located at the point �x. Density ρ(x, y, z) = ρ(�x) mass per unit volume located at the point �x. a point ρ(x, y, z) = ρ(�x). Energy Density u(x, y, z) = u(�x) energy per unit volume located at the point �x. The examples above are spatially dependent densities of “extensive quantities.” There are also spatially variable intensive quantities: Temperature T (x, y, z) = T (�x) is the temperature which would be measured at the point �x. Pressure P (x, y, z) = P (�x) is the pressure which would be measured at the point �x. Chemical Potential µi(x, y, z) = µi(�x) is the chemical potential of the species i which would be measured at the point �x. Each example is a scalar function of space—that is, the function associates a scalar with each point in space. A topographical map is a familiar example of a graphical illustration of a scalar function (altitude) as a function of location (latitude and longitude). qcqckHow Confusion Can Develop in Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . However, there are many other types of scalar functions of several arguments, such as the state function: U = U(S, V, Ni) or P = P (V, T, Ni). It is sometimes useful to think of these types of functions a scalar functions of a “point” in a thermodynamics space.MIT 3.016 Fall 2005 � W.C Carter Lecture 12 c73 However, this is often a source of confusion: notice that the internal energy is used in two different contexts above. One context is the value of the energy, say 128.2 Joules. The other context is the function U(S, V, Ni). While the two symbols are identical, their meanings are quite different. The root of the confusion lurks in the question, “What are the variables of U?” Suppose that there is only one (independent) chemical species, then U(·) has three variables, such as U(S, V, N). “But what if S(T, P, µ), V (T, P, µ), and N(T, P, µ) are known functions, what are the variables of U?” The answer is, they are any three independent variables, one could write U(T, P, µ) = U(S(T, P, µ), V (T, P , µ), N(T, P, µ )) and there are six other natural choices. The trouble arises when variations of a function like U are queried—then the variables that are varying must be specified. For this reason, it is either a good idea to keep the functional form explicit in thermody-namics, i.e., ∂U(S, V, N) ∂U(S, V, N) ∂U(S, V, N)dU(S, V, N) = dS + dV + dN ∂S ∂V ∂N (12-6)∂U(T, P, µ) ∂U(T, P, µ) ∂U(T, P, µ)dU(T, P, µ) = dT + dV + dµ∂T ∂V ∂µ or use, the common thermodynamic notation, � ∂U � � ∂U � � ∂U �dU = dS + dV + dN ∂S ∂V ∂N V,N S,N S,V (12-7)� ∂U � � ∂U � � ∂U �dU = dT + dP +