Week 8: Partial derivatives with constrained variables, Taylor's formula for two variables
Thomas Calculus, 11th edition, pages: 1049 - 1060
Sec 14.9 Partial Derivatives with Constrained Variables
In finding partial derivatives of functions like w = f(x,y), we have assumed x and y to be independent. In many applications, however, this is not the case. For example, the internal energy U of a gas may be expressed as a function U= f (P, V, T) of pressure P, volume V, and temperature T. If the individual molecules of the gas do not interact, however, P, V, and T obey (and are constrained by) the ideal gas law
PV = nRT (n and R constant),
and fail to be independent. In this section we learn how to find partial derivatives in situations
like this, which you may encounter in studying economics, engineering, or physics.
Sec 14.10 Talor's Formula for Two Variables
This section uses Taylor’s formula to derive the Second Derivative Test for local extreme values (Section 14.7) and the error formula for linearizations of functions of two independent variables (Section 14.6). The use of Taylor’s formula in these derivations leads to an extension of the formula that provides polynomial approximations of all orders for functions of two independent variables.
- Week 1: Three-Dimensional Coordinate Systems, Vectors, The Dot Product.
- Week 2: The Cross Product, Lines and Planes in Space, Cylinders and Quadric Surfaces
- Week 3: Vector functions, Modeling projectile motion, Arc length and the unit tangent vector T.
- Week 4: Curvature and the unit normal vector N, Torsion and the unit binormal vector B, Planetary motion and satellites
- Week 5: Functions of several variables, Limits and continuity in higher dimensions, Partial derivatives,
- Week 6: The chain rule, Directional derivatives and gradient vectors, Tangent planes and differentials
- Week 7: Extreme values and Saddle points, Lagrange multipliers,
- Week 8: Partial derivatives with constrained variables, Taylor's formula for two variables
- Week 9: Double integrals, Areas, Moments and centers of mass, Double integrals in polar form
- Week 10: Mid Term, December 14 - 18,2020 Week 11: Winter Break, December 21 - 24, 2020
- Week 12: Triple integrals in rectangular coordinates, Masses and moments in three dimensions, Triple integrals in cylindrical and spherical coordinates.
- Week 13: Substitution in multiple integrals, Line integrals, Vector fields, Work, Circulation and Flux,
- Week 14: Path independence, Potential functions and conservative fields, Green's theorem in the plane
- Week 15: Surface area and surface Integrals, Parameterized surfaces.
- Week 16: Stokes' Theorem, The Divergence theorem and a unified theory
- Chapters 15
- Department Chemistry
- Teacher
Uzma Ahmad
