Fall 2019  Math E21a weekly topics and assignments
Assignments will be updated as the course proceeds. Check back each week after class.
[last updated Friday, June 28, 2019 8:01 AM]
Date  Topics  Homework assignments (4th Ed.) [Solutions][underlined = graded]  
Thurs, Sept 5 Class #1 
Introduction to R^{2} and R^{3}; points vs. vectors; sum of vectors, scalar multiplication; difference vector; length of a vector; distance between points; unit vectors. Equations for circles and spheres; vector and parametric equations of a line in R^{2} and in R^{3}. 


Thurs, Sept 12 Class #2 
Dot product in R^{2}, R^{3} and its algebraic and geometric properties. Scalar and vector projections. Equations of lines and planes in R^{3} and their intersections. The cross product in R^{3} and its algebraic and geometric properties, triple scalar product. Applications to distance, area, volume. 


Thurs, Sept 19 Class #3 
Cross product, triple scalar product, continued. Equations vs. parameterizations. Brief survey of functions, graphs, and surfaces. Vectorvalued functions  parameterized curves in R^{2} and R^{3}. Velocity vectors. 


Note: If you have access to mathematical software such as Mathematica or Matlab or Maple, you might try graphing some of the functions in section 9.6 and some of the curves in 10.110.2 on the computer.  
Thurs, Sept 26 Class #4 
Velocity and acceleration vectors (10.2 and 10.4); arclength, unit tangent vector, curvature, unit normal vector (10.3). Functions of several variables; graph of a function of two variables (11.1); partial derivatives (11.3). 


Thurs, Oct 3 Class #5 
Limits and continuity (11.2). Partial derivatives and differentiability (11.3); linear approximation; tangent plane to the graph of a function of two variables (11.4). Rate of change of a function along a parameterized curve, the basic Chain Rule, directional derivatives and the gradient vector (11.5 and 11.6); gradients and normal vectors (11.6). Practice Exam #1 Solutions 


Thurs, Oct 10 Class #6 
Gradients and normal vectors (11.6); General Chain Rule (11.5); implicit differentiation (11.5); higher order partial derivatives (11.3); equality of mixed partial derivatives  Clairaut's Theorem (11.3). Midterm Exam 1 Solutions 


Thurs, Oct 17 Class #7 
Quadratic approximation (11.7); unconstrained optimization  finding maximum and minimum values of functions of two or more variables, 2nd Derivative Test for stationary points of a function of two variables (11.7); constrained optimization and the Method of Lagrange Multipliers (11.8). Extreme values of a continuous function defined on a bounded region. 


Thurs, Oct 24 Class #8 
Method of Lagrange Multipliers, continued; examples of unconstrained and constrained optimization in economics; Method of Least Squares; multiple constraints. Introduction to integration over regions in R^{2}. 
HW8: Read sections 11.8 and 12.1 (and maybe read ahead in 12.212.3). HW #8 problems (PDF)  due Thurs, Nov 1 (DL Nov 3) Note: Problems 2, 412 were chosen for grading 

Thurs, Oct 31 Class #9 
Multiple integrals in the calculation of area, volume, mass, population, and average value of a function. Iterated integrals; changing order of integration in an iterated integral; and the Fubini Theorem. Use of polar coordinates in calculating double integrals. Geometric center of a region (centroid) and center of mass. Triple integrals in Cartesian coordinates and calculation by successive slicing. 


Thurs, Nov 7 Class #10 
Triple integrals, continued. Applications of triple integrals: average value, centroid, mass, center of mass, weighted averages, moment of inertia. Triple integrals in cylindrical and spherical coordinates. 


Thurs, Nov 14 Class #11 
General change of variables in multiple integrals. Vector fields in R^{2} and R^{3}; line integrals and work done by a variable force along a parameterized curve. Calculation of line integrals. Fundamental Theorem of Line Integrals, independence of path, conservative vector fields. Test to determine if a vector field is conservative. Supplement on Conservative Vector Fields and the Fundamental Theorem of Line Integrals 


There's a good javabased tool for showing vector fields and flows in R^{2} at http://math.rice.edu/~dfield/dfpp.html. Choose the PPLANE option. You can enter new x and y component functions for the vector field or change the size of the window. To see a trajectory (flow), just click on a point in the phaseplane.  
Thurs, Nov 21 Class #12 
Fundamental Theorem of Line Integrals, independence of path, conservative vector fields, potential functions; Green's Theorem; divergence and curl of a vector field. Midterm Exam 2 Solutions 


No class on Thurs, Nov 28 due to Thanksgiving Holiday  
Thurs, Dec 5 Class #13 
Parameterized surfaces in R^{3}; integration on parameterized surfaces; surface area. Calculation of surface integrals; flux of a vector field through a surface. Geometric definitions of divergence and curl, and derivation of algebraic definitions from the geometric definitions. Statement of the Divergence Theorem and Stokes' Theorem. Supplement on surface integrals  toolkits for spheres, cylinders, graphs, and any parameterized surface. 


Thurs, Dec 12 Class #14 
Geometric definitions of divergence and curl, and derivation of algebraic definitions from the geometric definitions (time permitting). Sketch of proofs of the Divergence Theorem and Stokes' Theorem from these geometric definitions. Proof of Green’s Theorem from Stokes’ Theorem. Practice Final Exam Solutions No notes are permitted on the Final Exam. Calculators are OK, but only for ordinary calculations, i.e. you should actually carry out any necessary integrations. 


Thurs, Dec 19 8:00pm 
The Final Exam will take place on Thursday, December 19 from 8:00pm to 10:00pm in Emerson Hall 105 (tentative). This will be a 2hour exam. There will be no special arrangements for alternate times or locations for the Final Exam except for students with disabilities. Any medical exceptions must be requested directly via the Extension School. Distance students should make the necessary arrangements through the Exams Office. 