Course Information, Section 01, 10am

Course Information, Section 02, 11am

Class Log and Announcements

• Announcement: If you plan to take Differential Equation this summer, please register it as soon as possible. It is in danger of being canceled due to low enrollment. If you take Diff. Eq., some of the materials that we learned or skipped in our course will become easy for you to understand. Diff. Eq. is one of the few courses in higher level that are not theoretical.


• Information for the final exam as of 4/28


• Quiz 11 this Monday will be on everything covered since the last quiz.


• April 25:
  Section 16.4 Green`s Theorem.
  Examples 1, 2, 4
  Homework 9, 13, 15, 18


• April 24:
  Section 16.3 The fundamental theorem for line integrals.
  Fundamental Theorem for line integrals, (2) p.1074, is an important theorem, but practical to use when you can find f such that grad f=F for a given vector field F. What you need to know (for our course) from this theorem is: Independence of path when F is a conservative field.
  Skip: Differential equation type of materials such as Example 4(a), Example 5; Conservation of Energy
  
  Homework Section 16.3 Exercises 19, 21 (We did 20 and 22 in class. Please finish the computations of 20, 22.)


• April 23:
  Section 16.2 Line integrals of vector fields.
  By the equation in the box on p.1071 (we derive it in class), you can simplify evaluating line integrals of vector fields to line integrals with respect to x, y and z. Of course, you can still evaluate F(r(t))(dot)(r`(t)) directly if you want, though.
  Examples 7, 8
  Skip: Formal definition of line integrals of vector fields (11) and (12), p. 1069.
  
  Section 16.3 The fundamental theorem for line integrals.
  Conservative vector fields: For a given vector field F, it is not trivial at all to find a function whose gradient is F, as I mentioned in class. You need to know a math subject called ``differential equation``. The textbook briefly discusses it. But I think it is not meaningful but just a burden to students that we cover it so briefly and expect them to know how to solve differential equation type of problems. Once you take a course on differential equation, you will find what`s written in the textbook quite basic. What you should know is (6) in p.1078, the criteria to determine whether F is conservative.
  Examples 2, 3
  
  Homework Section 16.2 Exercise 21; Section 16.3 Exercises 5, 6 (Do only determining conservativeness. If you have taken a course on diff. eqn., try to fine f; it won`t be so difficult.)


• April 21:
  Section 16.1 Vector fields.
  Examples 1, 6
  Homework 17, 31


• Quiz 10 this Monday will be on evaluating line integrals in Section 16.2.
  Let me remind you of an announcement made at the beginning of the semester and written in the bottom of this Class log page:
  We can spend 10 minutes to discuss problems before the quiz for last minute questions if you have any. If you have long questions, it would be reasonable that you come and see me before the class. Please check out my office hours in the syllabus.


• April 18:
  Section 16.2 Line integrals.
  Examples 1, 2, 4, 5, 6 (We did more examples in class from exercises.)
  Skip: Formal definition of line integrals in terms of limits, p.1062. Mass and the center of mass p.1064--1065.
  Homework 3, 8, 10, 14


• April 17:
  Section 16.2 Line integrals. Study all the examples that we did in class today. I will assign homework from the exercises after tomorrow`s class.


• April 16: Review the followings:
  • 13.1 Vector functions and space curves
  • 13.2 Derivatives of vector functions
  • 13.3 Arc length functions, especially the formula ds=|r`(t)|dt and its expansion version in terms of dx/dt and dy/dt (and dz/dt for 3-dim case)
  • 12.5 Parametric equations and vector equations of lines; How to express a line segment by restricting the range of t
  • Parametrization of curves that are frequently seen; circles (2-dim) and circular helix (3-dim)
  Please review all of these, and feel comfortable with the language of vectors.


• April 14: Hour exam 3


• April 10:
  Section 15.9 Change of variables in multiple integrals. See yesterday`s homework.


• Information for the hour exam 3 as of 4/9
  
  • Hour exam 3 will be on Section 12.7 and Chapter 15. We will have Q&A session this Friday 4/11. Bring your own questions.


• April 9:
  Section 15.9 Change of variables in multiple integrals. There are basically two tasks: finding Jacobian of the transformation, and being able to see the preimage and the image of the transformation. The former is usually easy but the latter can be tricky.
  Examples 1, 2, 4
  
  Skip Theoretical background of obtaining Jacobian in pp.1042(bottom)--1043(right above the definition of Jacobian).
  Example 3 (This example is fairly difficult; you have to create an appropriate transformation from the problem!! Other than that, the area S in the uv-plane is not much simplified from the domain D in the xy-plane. The point of this example is that, since the function e^(x+y)/(x-y) does not have an immediate antiderivative w.r.t x or y, change the variables to find an antiderivative. You can think it as a substitution in double integrals. You will learn more on the topic if you try this example.)
  
  Homework 1, 5, 13, 17, 18 (The transformations of these problems are easier than Example 1 of the textbook in which you divide the borderline into segments and worked piece by piece. However, on these exercise problems, you can see the images and preimages immediately by plugging the transformations into the equation of the given surfaces.


• April 7: Today, a student pointed out a typo in the handout distributed last Friday. On the first page, the spherical equation for the half-cone should be phi=pi/4 (not rho=pi/4).
  Section 15.7/ 15.8 Triple integrals. Volume using triple integrals.
  Homework Do the following problems, all of which are related to (part of) a unit sphere.
  • Set up double integrals to find a volume of the upper hemisphere of radius 1, centered at the origin using (1) rectangular coordinates, and (2) polar coordinates.
  • Set up triple integrals to find a volume of the upper hemisphere of radius 1, centered at the origin using (1) rectangular coordinates and (2) sphere coordinates. Also try in cylindrical coordinates, which we didn`t do in class formally but it`s not difficult; you will realize it is ``essentially`` polar coordinates of double integrals.
  • Set up double integrals to find the surface area of the upper hemisphere of radius 1, centered at the origin using (1) rectangular coordinates, and (2) polar coordinates.


• Quiz 9 this Monday will be on Sections 12.7, 15.6, and 15.7/15.8(on the parts that we have covered).


• April 4:
  Section 15.7/ 15.8 Triple integrals. dV=r dz dr d(theta) in cylindrical, dV=rho^2 sin(phi) d(rho) d(theta) d(phi)
  We have not finished these sections yet. We will find volumes of 3-dim solids using triple integrals on Monday.
  
  Examples 15.7. 1, 2
  Examples 15.8. 2, 3(For 10am section only, we will finish the example in spherical coordinates that is similar to the example 3 on Monday.)
  
  Skip 15.7 The formal definition of triple integrals via limits. Mass, and moments of inertia. Probability (All of these are ``obvious`` generalizations of double integrals as you can see in other computations of triple integrals.)
  Homework Study the handout that was distributed during class. More precisely, (1) memorize the equations of the surfaces on the first page, (2) try to obtain all the inequalities that describe the solids on the second and the third pages on your own, and understand why one coordinate system is better than others depending on (integration domain) solids.
  Exercises 15.7. 12
  Exercises 15.8. 5, 6, 11, 17, 21, 33, 35 (For 10am section only, you may postpone the exercises in spherical coordinates.)


• April 3:
  Section 12.7. Cylindrical and Spherical coordinates.
  Rectangular coordinates vs. Spherical coordinates; You are expected to identify spheres and half-cone from given cylindrical equations. Remember that you should memorize equations that convert coordinate systems: rectangular <-> cylindrical (1),(2), p.827, and rectangular <-> spherical (3), (4), p.829.
  Example 4
  
  Homework Exercises 15, 31, 35, 37, 60


• April 2:
  Section 12.7. Cylindrical and Spherical coordinates.
  Rectangular coordinates vs. Cylindrical coordinates; You are expected to identify cylinders and half-cone from given cylindrical equations. Also, you should be able to convert those equations from rect. coor. to cylind. coord., and vice versa.
  Examples 1, 2
  
  Homework Exercises 3, 7, 9


• Mar. 31:
  Section 15.6. Surface area
Examples 1--2

Skip: From 15.6, you only need to know the double integral formula for the surface area (formulas (2) and (3) in pp.1020--1021); these are the formulas needed to compute surface area in practice. You can skip the formal definition of surface area via limits (formula (1), p. 1020).

Homework Exercises 7, 9, 10


• Quiz 8 this Monday will be on Sections 15.4--15.5, and average values.


• Mar. 26:
  Section 15.5. Applications of double integrals.
  Average values of f(x,y) (This is from Section 15.2. I think I said it was from Section 15.1 during class. Please correct it to Section 15.2.); Moments and Centers of mass (Centroid)
  Example 2
  
  Homework 15.2 Exercises 33; 15.5 Exercises 3, 8, 11, 14


• Mar. 26:
  Section 15.4. Double integrals in polar coordinates
  Examples 1--3
  Homework Exercises 1, 3, 5, 7, 9, 17, 21


• Mar. 24:
  Section 15.4. Double integrals in polar coordinates There is no homework from exercise problems. Just look through the note. Tips to learn this subject:
  (1) Keep in mind the meaning of polar coordinates and the directions of each parameter r and theta when you describe a region D in the inequalities of r and theta;
  (2) It is a lot better to remember replacing dA=r dr d(theta) in polar coordinate to compute double integral SS_D f dA rather than directly jumping into memorizing double integrals in polar coordinates.


• Quiz 7 this Monday will be on Sections 15.2--15.3.


• Mar. 13--14:
  Section 15.3. Double integrals over general regions Remember that you should know how to sketch regions and give the relevant inequalities to set up the double integrals.
  Examples 1--5. Example done in class related to formula (10) on p.1001;
  Among properties of double integrals, you need to know only (10) on p.1001, the formula to compute the area A(D) of a region D by a double integral. This formula is a must-know and will be important later on. But unfortunately, there is no example for this formula in section 15.3 of the textbook. So make sure to look through the example that we did in class.
  Skip: You don`t need to memorize the formal properties of double integrals given in pp.1000-1001, except for formula (10).
  
  Homework Exercises 3, 7, 11, 17, 19, 21, 41, 43


• Mar. 12:
  Section 15.2. Iterated integrals Geometric idea behind the iterated integrals.
  Today when I gave a lecture, I wrote the title as ``...idea behind the double integrals...``. But the ``iterated integrals`` would be more appropriate (since that was what we did today) than the ``double integrals``. Please make the changes for your note. The double integrals have various ways to interpret and we will learn them soon.
  Examples 1--5; We didn`t do Example 5 in class. But make sure you read it through on your own.
  
  Homework Exercises 5, 13, 18, 25


• Mar. 10: Hour exam 2


• Mar. 7: Review for hour exam 2


• Mar. 6:
  Section 15.2. Iterated integrals. Definitions and computations.


• Mar. 5: Information for the hour exam 2 as of 3/5
  
  • Hour exam 2 will be on Chapter 14 and we have covered all the material. We will have Q&A session this Friday 3/7. Bring your own questions.
  • By demand and to lighten your load of work, you need to know only Lagrange multipliers method for the optimization problem on the exam.
  • Read the lecture note and this webpage more carefully. I always write down how the problems is going to be asked in these two (and specify it in class as well).


• Mar. 5:
  Section 14.8. Lagrange multipliers. On exams, you should give the set-up -- Max/Min f, Subject to g=k -- clearly; while computing, the four equations of the system (three equations from the equality of grad f(x,y,z) = lambda g(x,y,z) together with g(x,y,z)=k ) should be clear.
  Examples 1, 4.
  Skip: Two constraints
  
  Homework Exercises 8, 28, 29. As I mentioned in class, the Lagrange multiplier method work for two variable functions, in which case there are fewer number of equations and it is simpler to compute. Example 2 and Exercises 3 and 5 would be good to look at to practice your computation skill at an easy level.


• Mar. 3:
  Section 14.8. Lagrange multipliers. When f(x,y,z) is a function to optimize and g(x,y,z)=k is a given restriction, the points (x,y,z) that give optimum values occur when grad f= c grad g for some non-zero constant c (in the textbook, they use *lambda* instead of c but we cannot write lambda here so I write c instead).


• Quiz 6 this Monday will be on Sections 14.6 and 14.7.


• Feb. 29:
  Section 14.7. Maximum and minimum values. Optimization problems.
  Examples 3, 6
  For optimization problems on quizzes or exams, you are expected to give the function to optimize and the restriction in the way that we did in class: clearly write two set-ups, one with three variables and the other with two variables.
  
  Homework Exercises 1(a)(b), 5, 7, 43


• Feb. 28:
  Section 14.6. The gradient vector and the directional derivatives.
  Relation between the gradient vector of f and its level curves/level surfaces, tangent vectors/ tangent lines
  Examples 8(tangent plane only), Make sure to study the examples done in class
  Skip:The definition of the directional derivatives using limits. Normal lines on p.948
  
  Section 14.7. Maximum and minimum values.
  Local maxima, local minima, or saddle point for two-variable functions f(x,y) using second derivative test.
  Examples 1, 2
  
  Homework for section 14.6 39, 43 (For 39 and 43, do only (a)tangent planes), 52(Hint: use the following facts: (1) two planes are parallel if and only if their normal vectors are parallel; and (2) two (non-zero) vectors a and b are parallel if and only if one is a multiple of the other, i.e., a=cb for some constant c.)


• Feb. 27:
  Section 14.6. The gradient vector and the directional derivatives.
  Geometric meaning of the gradient vector: grad f is the vector in the xy-plane that points in the direction of the largest increase} (or steepest direction) of f and whose magnitude is equal to the rate of increase in this direction. Be aware of three ``different`` expressions that are basically the ``same``, which we discussed in class.
  Examples 6, 7 exercise 33 (we saw this problem as an example in class): Some of the examples/problems that we did in class ask questions in a slightly different way from the way in those examples of the text. Make sure to know all of them!
  
  Homework: Exercises 21, 34(c)* (Hint for the third question (about the angle): the tangent of the angle of climb is the rate of change of the elevation, expressed as the feet gained in elevation per foot gained in horizontal distance)


• #11 has been added to the homework list on Feb. 25.


• Feb. 25:
  Section 14.6. The gradient vector and the directional derivatives.
  Geometric meaning of directional derivatives in the direction of a unit vector u: D_uf(x_0,y_0) is the slope of the surface given by a function z=f(x,y) at the point (x_0,y_0) in the direction of the vector u. It`s the same as the rate of change of f at (x_0,y_0) in the direction of u. Note that if u=i, the directional derivative D_if(x_0,y_0) is easily computed (through the dot product of gradient and i) into f_x(x_0,y_0), as expected. Similarly, if u=j, D_jf(x_0,y_0)= f_y(x_0,y_0).
  Example 4
  
  Homework: Exercise 11, 31(a)


• Quiz 5 will be on Sections 14.3(covered from 2/18 onwards), 14.4 and 14.5 this coming Monday.


• Feb. 22:
  Section 14.6. The gradient vector and the directional derivatives.
  Definitions (in fact, formulas) of the gradient vector and directional derivatives (8) and (9) on p.944, and (13) and (14) p.945. (The formal definitions of directional derivatives are through limits as in (2) on p.941, but we skip these formal definitions.) Keep in mind that, the vector u in the notation of directional derivative D_uf must be a unit vector; Problems can ask the directional derivatives in the direction of any vector v that is not a unit vector (see examples 4, 5(b)), then you have to find a unit vector u=v/|v| that has the same direction as v.
  
  Examples 2 (We didn`t do this example in class but please do it on your own. The phrase ``the unit vector given by an angle pi/6`` means nothing but u=cos(pi/6) i + sin(pi/6) j. Note that this vector u is a unit vector. The solution given in the text book may look different than the way we did in class, but they are essentially the same: Set the directional derivative D_uf = grad f dot u with u=cos(pi/6) i + sin(pi/6) as we did in class. Then you can see the rest is the same as the solution in the text), 3, 4, 5
  
  Homework: Exercises 4(Do Example 2 first), 7
• 
  
• Feb. 21: I couldn`t finish Section 14.5 (more precisely, implicit differentiation) today and will finish it tomorrow.
  Section 14.5. The chain rule.
  The chain rule works nicely through Tree Diagram. Note that the derivatives of z in a parameter resemble its differentials.
  
  The formulas 6 and 7 on p.939 give us convenient implicit differentiations via F. To make use of those formulas, you should know how to relate a given equation (e.g., x^3+y^3+z^3+6xyz=1) to an appropriate F (e.g., F(x,y)= x^3+y^3+z^3+6xyz or F(x,y)= x^3+y^3+z^3+6xyz-1). Compare Example 9 of 14.5 to Example 4 of 14.3.
  Note that the given equation x^3+y^3+z^3+6xyz=1 in the example can be interpreted as the level surface of F(x,y)= x^3+y^3+z^3+6xyz at the level of 1, or the level surface of F(x,y)= x^3+y^3+z^3+6xyz-1 at the level of 0; Either choice of F will work. However, the former F is more typical choice.
  Examples 1, 2, 3, 4, 5
  
  Homework: Compare Example 9 of 14.5 to Example 4 of 14.3; Exercises 1, 5, 7, 21, 28, 31, 43*(note that the given derivatives are squares of first partial derivatives, not second partial derivatives)


• Feb. 20:
  Section 14.4. Tangent lines, Differentials.
  Examples 1, 4 (Note that we modify the questions of Example 4 in class.)
  Skip Theoretical background how to obtain the equation of the tangent planes (though we covered in class and it`s good to know), Differentiable (p.926), Linear Approximation (Linear approximation is just an simple application of tangent lines. At this stage, it may look difficult. If you are able to understand tangent line better later on, you will realize there is no new math in linear approximations.)
  
  Homework: Exercises 1, 23, 25, 30(Do only dz), 36(Hint: Think P,V,T as z,y,x, respectively.)


• Feb. 18:
  Section 14.3. Partial derivatives. On this webpage, I can`t type the notation for partial derivatives (the rotated ``e``), and I will use ``e`` instead, such as ef/ex.
  Remark: The variable z in x^3+y^3+z_3+z+6xyz=1 is a function of variables x,y hence we need the implicit differentiation to find the partial derivative ez/ex. On the other hand, z in the function f(x,y,z)=x^+y^3+z^3+z+6xyz is an independent variable from x and y. Hence, when you find ef/ex, z should be treated as constant so that ez/ex=0. However, they are related to each other: setting f(x,y,z)=1 yields the equation x^3+y^3+z_3+z+6xyz=1, hence x^3+y^3+z_3+z+6xyz=1 can be read as the level surface of f(x,y,z) at the level of 1. This view point will be useful later in 14.6.
  Examples 4, 5, 6
  Skip: The limit definition of partial derivatives. Partial differential equations. The Cobb-Douglas production function.
  Homework: Exercises 31, 41, 47


• Quiz 4 will be on Sections 14.1 and 14.3 this coming Monday. Though there is no homework on some particular sections (like 14.1), you are still expected to know what we covered in class and they can be on exams and quizzes. In case some materials are not on exams, I always specifically write them on this course webpage.


• Feb. 15:
  Section 14.3. Partial derivatives. To learn partial derivatives smoothly, you need a basic geometric intuition in partial derivatives that we did in class.
  Comment: The notations f '(x,y) or df(x,y)/dx do NOT make any sense for a function of two variables. It is essential to use appropriate notations in exams or quizzes: d for the derivatives of functions of one variable; the 180 degree rotated e (called ``round``) for the partial derivatives of functions of more than one variables.
  Examples: 1, 2, 3
  
  Homework: Exercises 13, 35, 37, 45(a), 46(b) (note that in 45 and 46, the functions f and g are ordinary one-variable functions so they have well-defined derivatives f' and g', which are df/fx and dg/dy, respectively).


• Feb. 14:
  Section 14.1. Functions of several variables.
  Domain, Level curves/Level surfaces. Note that the typical task (2) that we did in class does not occur among the book examples but will come up later. Make sure to know how to solve that type of problems.
  Examples 4,5,10,11, 15; Figures 11, 20.
  
  Homework: Section 14.1


• Feb. 13: The graded exam was distributed.
  The tangential and normal components of acceleration were derived.


• Feb. 11: Hour Exam 1


• There was a typo in the list of homework for Section 13.4. Add #20 and remove #30.


• Information for the hour exam 1 as of 2/6


• Feb. 7:
  Section 13.4. Velocity and acceleration.
  Physical interpretation of vector functions: position, velocity, speed, acceleration
  The Newton's Second Law: F(t)= ma(t), a fundamental law of physics, where F(t) is the force acting on an object of mass m at time t, and a(t) the acceleration caused by this force at time t.
  Normal and tangential components of the acceleration : The acceleration vector a = a(t) can be decomposed into a tangential component a_T and a normal component a_N. These components can be composed via the two pairs of formulas (see (8), (9) and (10) on p.875). The formula (8) are useful if you are given the speed v and curvatures. The formulas (9) for a_T and a_N are almost same except for the dot product and cross product.
  Examples 1--4, 7
  Skip: Kepler's law
  
  Homework: Section 13.4 Exercises 3, 5 (you can omit the sketch in these problems), 15, 20, 33(This was assigned as homework during class), 38*(For the purpose of the first exam this Monday, you can skip this problem. It`s an interesting problem though.)


• Feb. 6: I forgot to distribute the solution of Quiz 3 today in class. I will distribute it tomorrow. Meanwhile, they are ready and hung on the door of my office; you can pick up one copy.
  
  Section 13.3. Arc length and curvature. The curvature measures how curved the curve is at a point. For a general space curve, the curvature usually varies from point to point. It can be computed using one of the two formulas (9) on p.864 and (10) on p.865. You should know both of these formulas.
  You should also know two special cases:
  1. Curvature of a line: A straight line has curvature 0, as one would expect. (Exercise: Show that this follows from the formulas above and the vector equation of a line, as briefly mentioned in class.)
  2. Curvature of a circle of radius R: It's a constant 1/R (i.e., the reciprocal of the radius). This is easy to remember and you should know this, without having to use the formula for the curvature.
  
  Unit tangent vector T(t), normal vector N(t), and binormal vector B(t) of the curve C given by r(t) at a the point t are unit vectors that are mutually orthogonal at t on C. You should know how compute them algebraically and the reason why are mutually orthogonal. It`s clear that T and N are unit vectors from the definitions. It`s an exercise to find out the reason why B is also a unit vector. (In general, the cross product of two unit vectors that are mutually orthogonal is also unit, while the cross product of two unit vectors is not necessarily a unit vector.)
  The normal plane of a curve C at t determined by N(t) and B(t) is perpendicular to T(t); the osculating plane of a curve C at t determined by T(t) and N(t) is perpendicular to B(t).
  
  Examples 1, 3, 6, 7
  Skip: Parametrizing a curve w.r.t. arc length (p.863 bottom and Example.2); the definition of curvature (8),p.864; formula for curvature of a curve $y=f(x)$ (p.866, (11), and Example 5); osculating circles (bottom of p.867, Example 8.)
  
  Homework: Section 13.3 HW mentioned during class; the exercises mentioned above; Exercises 17, 20, 39, 41, 45*


• Feb. 4: I have updated this course webpage a little bit; adding examples to study, materials to skip at each sections as well as spacing between lines.


• Feb. 4:
  Section 13.2. Derivatives of vector functions.
  Examples 2, 3, 5(Important!!!), 6
  Skip: Smooth curves, cusps (top of p.859)
  
  Section 13. 3. Arc length and curvature. Arc length.
  Example 1
  
  Homework: Section 13.2 Example 6; Exercises 23, 33, 39, 47*, 49*.
  Section 13.3 Exercise 1


• Quiz 3 will be on Sections 12.5 and 13.2 this coming Monday.


• Feb. 1:
  Section 12.6. Quadratic surfaces. We know from 12.5 that an equation like 4x+3y+5z=1, in which x,y, and z each occur as linear functions (as opposed to quadratic or more complicated functions) describes a plane. Equations in which the variables x,y, and z each occur as quadratic form (such as x^2+y^2+z^2=a^2) represent quadratic surfaces. A full list is given on p.836, but you need not memorize this table, or any general formulas for quadratic surfaces (and there will be no hw/quiz/exam problems from this section). Though, it might be a good idea to draw two quadratic surfaces x^2+y^2=a (cylinder) and x^2+y^2+z^2=a^2 (sphere) to improve your intuition. A typical strategy to sketch such surfaces is by ``layers``, as briefly mentioned in class. There are nice pictures explaining the idea in textbook.
  
  Section 12.7. Cylindrical and Spherical coordinates. We will cover this section before Chapter 15 later since the material in this section will not be needed until we get to multiple integrals in Chapter 15.
  
  Section 13.1. Vector functions and space curves. The main purpose of this section is to introduce the concept of a vector function and its geometric interpretation as a space curve. You won't be asked to draw space curves. Of the special curves given here, only helix is important, and you should be able to recognize a helix from its equation. There will be no problems on limits.
  Example 4
  
  Section 13.2. Derivatives of vector functions.
  Example 1
  
  Homework: Section 12.5 Example 10 (mentioned in class).
  Section 13.2 Study the second derivative r``(t) that is defined in a similar manner as the first derivative (see bottom of p. 858).; Exercises 11, 17, 19, 21


• Jan. 31:
  Section 12.5. Equations of lines and planes.
  Typical tasks:
  • Finding the equation of a line
  • Finding the equation of a plane
  • Finding the relations between lines, between a line and a plane, or between planes. If intersect, find the equation of the intersections (it could be a point or a line.) In case of planes, find the angle between them, which is defined as the acute angle between two normal vectors.
  • Finding the distance from a point P_1 to a plane. (We will finish this tomorrow.) The procedure of finding a distance is a brilliant application of vector projections and scalar projections in Section 12.3. Understand the procedure, rather than memorizing the formula (9) given in p. 828.
  
  Examples 3, 6 -- 10
  
  Homework: Section 12.5 Exercises 3, 5, 7, 13, 19, 27, 31, 39, 46, 50 ,65*, 67* (Some of them overlap from yesterday`s. You can do the distance problems after tomorrow`s class.)


• Jan. 30:
  Section 12.5. Equations of lines and planes. The vector equation , the scalar equation, and the linear equation of a plane.
  Examples 4, 5
  
  Homework: Section 12.5 Exercises 3, 7, 31


• There is a slight change of office hours. Please check out the Course Information above.


• Jan. 28:
  Section 12.5. Equations of lines and planes. The vector equation and the parametric equations of a line.
  It`s not difficult to find the equations of a line as long as memorize the formulae (especially parametric equations) in case of simple problems. However, understand the topic in the language of vectors geometrically so that advanced problems that we will do this Wednesday can make sense to you.
  Examples 1, 2(parametric equations only)
  Skip: symmetric equation of a line, direction numbers (the bottom of p.823-- the top of p.824)
  
  Homework: Section 12.5 Example 2


• Quiz 2 will be on Section 12.4 this coming Monday. Possible Q&A time for the first 10 minutes, followed by quiz.


• Jan. 25:
  Section 12.4. The cross product, finished.
  Skip: You need not memorize the last(6th) of the properties listed in the box on p.818 ; this is, by far, the most complicated and least intuitive formula in the list, and also the least important. (The other properties, however, are important, but are also easy to remember.)
  
  Homework: Section 12.4 HW mentioned during lecture today; Example 3; Exercises 16, 23, 25, 29, 45*(*Non-routine problems; won't be on quizzes, but may be on exams.)


• Jan. 23:
  Section 12.4. The cross product. The cross (or vector) product of two vectors a and b produces another vector. It can be defined algebraically ( as a determinant of a certain 3 by 3 matrix) and geometrically (as a vector that is orthogonal to both a and b with magnitude).
  
  The vector product has many applications in physics and mathematics. In particular, it can be used to check if two vectors are parallel, compute the area of a parallelogram determined by two vectors, and find vector that is orthogonal to two given vectors, or a vector that is orthogonal to the plane determined by three points. An application involving both the dot and the vector product is the computation of a volume of a parallelepiped, via a formula called the scalar triple product. (We did some of these applications today, and will do the rest tomorrow.)
  
  In contrast to the dot product, the vector product makes only sense for 3-dimensional vectors. Also, the rules for the vector product (listed on p.818) are somewhat different from those of dot product. In particular, the order of the vectors in the product matters: changing the order changes the resulting vector into its negative( the same vector going to the opposite direction).
  
  Examples 1, 2, 3, 4, 5
  
  Homework: Section 12.4 HW mentioned during lecture today; Exercises 8, 9(a)(b)(c)


• Hint for exercise 28 in section 12.3: Put u= (x,y) as a vector that we are looking for. From the problem, u has two properties; first, unit vector; second, the angle 60 degrees with the given vector v. Use the algebraic form of the magnitude of a vector to express |u|= 1 in terms of x and y; Rewrite the geometric definition of the dot product of u and v in terms of x and y. Now you have two equations in variables x and y. Solve them!!!


• Quiz 1 will be on Sections 12.1--12.3 this coming Wednesday. We will have quiz for 10 minutes at the beginning of class. We can spend 10 minutes to discuss problems before the quiz for last minute questions if you have any. If you have long questions, it would be reasonable that you come and see me before the class. Please check out my office hours in the syllabus.


• Jan. 18:
  Section 12.3. The dot product, finished. As mentioned in class, you need to know how to obtain scalar projections and vector projections as well as memorizing the final forms of those.
  Example 6
  Skip : You can skip the paragraph on direction angles(p.809 bottom - 810 top), and the physical applications given in Examples 7 and 8 of 12.3.
  
  Section 12.4. The cross product.
  
  Homework: Section 12.3 Exercises 36, 39, 41, 42, 49*, 50, 51*, 57*. Section 12.4 Exercises 13, 17
  (* Problems with * will not be on quizzes, but may be on exams.)


• Jan. 17:
  Section 12.3. The dot product. In 12.3 and 12.4, two ways of multiplying a vector with another vector are introduced. The first, called the dot product, produces a scalar; the second, called the cross product produces another vector. Both products can be computed algebraically in terms of the components of the vectors, and geometrically in terms of the magnitude and directions of the vectors. While the cross product makes only sense for 3-dimensional vectors, the dot product can be defined for vectors in any dimension.
  Always indicate which type of multiplication of vectors you mean: a centered dot for the dot product, and ``times`` symbol (x) for the cross product. A dot should not be used when multiplying a scalar with a vector (as in ``ca``), or when multiplying two scalars. Also, remember to put arrows on quantities that denote vectors (which on this webpage, and in the book, are typeset in boldface).
  Examples 1, 2, 3, 4, 5
  
  Homework: Section 12.3 HW mentioned during lecture today; Exercises 1(a)(e), 3, 9, 13(a), 23(a)(c), 28*
  (*Problems marked by an asterisk are non-routine. Those type of problems will not be on quizzes, though it is fair to put them on hour exams.)


• Jan. 16: Announcement: The first quiz will be held next Wednesday 1/23. It will be on Sections 12.1--12.3.
  
  Section 12.2. Vectors, finished.
  Examples 1--6
  
  Homework: Section 12.2 HW mentioned during lecture today; Read the bottom six lines in p.803; Examples 4,6; Exercises 6(b)(e)(f), 21, 23, 25, 26


• Jan. 14: Announcement: We didn`t have a quiz today and can`t take a quiz this coming Monday 1/21, Martin Luther King Day. Hence we will have the first quiz either this Friday 1/18 or next Wednesday 1/23. We will decide the date by majority this Wednesday in class.
  
  Section 12.2. Vectors. It is essential that vectors be distinguished from scalars (i.e., ordinary numbers) through appropriate notation. In the book and on this web page, vectors are denoted by boldface letters, while scalars are set in ordinary typeface. Since we can`t use boldface in handwriting, we will indicate a vector by placing an arrow above the letter. You should follow this notational convention in your quizzes and exams, and always denote vectors by an arrow above the letter. You will get points deducted if you don`t adhere to this convention.
  
  Homework: Section 12.2 Examples 1, 3, 4(only |a| and a+b); Exercises 6(a)(c)(d), 7, 9, 17(only |a| and a+b), 19(only |a| and a+b)


• There is no quiz this coming Monday 1/14.


• Jan. 11:
  Section 12.1. 3-dimensional coordinate systems, finished. A geometric intuition is important in this course. To improve that, try to obtain all the examples in the section for yourself, as we were slow at this section. I will be much faster from next section.
  Examples 1 -- 6
  
  Homework: Section 12.1 Exercises 10(a)(c)(e), 15, 20, 23, 31


• Jan. 10:
  Section 12.1. 3-dimensional coordinate systems. This section is largely a review of known material. I did not finish this section, and will finish the section tomorrow.
  
  Homework(not to be turned in): Section 12.1 Exercises 1


• Jan. 9: Course introduction