Review: Functions and Graphs and Review: Functions and Graphs - Solution
You are expected to remember these graphs and should know how to read/interpret their behaviors.
Section 12.1. 3-d Coordinate system and Surfaces ; This section is largely a review of known material.
Quiz 1 this Thurs 8/22 will be based on the topics covered this Monday, the first day of the class. You will practice similar questions during the recitation before taking the quiz,
Recitation and Quiz 1
Traces/Cross-sections and Constructing 3-d Surfaces (Examples in Section 12.6)
Section 12.2
Geometric definition of vectors; Arrow
12.2 A vector is determined by two factors: (i) Magnitude (length of the arrow), and (ii) Direction.
A vector is determined by two factors of an arrow: (1) length, and (2) direction. Two vectors are said the same if and only if they have the same length and the same direction.
12.2 Algebraic expressions of vectors; Components and Notations
Components = Terminal minus Initial (informally speaking)
Notations on 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.
Don't mix (a_1,a_2,a_3) and < a_1,a_2,a_3 >. The former with parentheses is a coordinate of a point that we are used to. The latter with brackets is a component of a vector. One way to connect them is that < a_1,a_2,a_3 > is the position vector of the point (a_1,a_2,a_3).
You should follow this notational convention in your quizzes and exams. You will get points deducted if you don`t adhere to this convention.
WebAssign: HW 1 Due 8/28 Wed 11:59pm
You will be able to do #1--#9 with the classes covered until 8/23 Fri.
For #10 & #11, you can wait until after 8/26 Mon. class.
Quiz 2 on Thurs 8/29 will be based on the materials that correspond to Homework 1.
Recitation and Quiz 2
Sections 12.3 and 12.4. The dot product and The cross product
In 12.3 and 12.4 two ways of multiplying a vector with another vector are introduced.
-The dot (or scalar) product of two vectors a and b produces a scalar;
- 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 cross product has many applications
Both products can be computed
-algebraically in terms of the components of the vectors, and
- geometrically in terms of magnitudes and directions of the vectors.
While the vector (or cross) product makes only sense for 3-dimensional vectors, the dot product can be defined for vectors in any dimension.
Geometric Applications of the dot product
Determine whether two vectors are orthogonal (i.e., perpendicular)
Compute the angle between two vectors
Compute the projection of a vector onto another
Scalar projection (comp_a b) ; this is a scalar
Vector projection (proj_a b) ; this is a vector
Geometric Applications of the cross product
Determine whether two vectors are parallel
Compute the area of parallelogram determined by two vectors
Applications involving both the dot and the cross product are
the computation of a volume of a parallelepiped, via a formula called the scalar triple product.
Determine three vectors are coplanar.
In-class exercises for Geometric Applications of the cross product and the scalar triple product.
Always be mindful if you are dealing with vector or scalar.
Comment on notation: 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).
WebAssign: HW 2 Due 9/4 Wed 11:59pm
Textbook Exercises 12.3 #45, #46 (Click to see the exercise questions)
It's about the vector projection and the orthogonal projection.
You can find them in the eBook. They are good exercises but not available on WebAssign.
Not required to submit these exercises but you should do them to be better prepared for the upcoming materials & quiz and future exams.
Quiz 3 on Thurs 9/5 will be based on the materials that correspond to Homework 2.
Recitation and Quiz 3
Sections 12.3 and 12.4
Warpped up Applications involving both the dot and the cross product
Briefly discuss some of the questions in In-class exercises for Geometric Applications of the cross product and the scalar triple product.
Section 12.5. Lines and Planes
12.5 Lines; Formulas and Examples
Skip: You may skip symmetric equations of lines.
12.5 Planes: Formulas and Examples and 12.5 Planes: Formulas and Examples - full (solutions included)
12.5 Relations among lines and/or planes and 12.5 Relations among lines and/or planes - full (Example 3 solutions included)
There can be hundreds of different ways to ask these questions, and it's not possible to have a fixed method to apply all the cases. You should solve them case by case; think of geometric figures and then often use school algebras to fill the details.
WebAssign: HW 3 Due 9/11 Wed 11:59pm
You will be able to do #1--#5 without a problem with the classes covered until 9/6 Fri.
For #6-#11, you can wait until after 9/9 Mon. class.
Quiz 4 on Thurs 9/12 will be based on the materials that correspond to Homework 3.
Recitation and Quiz 4
Section 13.1 Vector Functions and space curves
13.1 A vector function (or vector valued-function) and Space curve
With the image vectors placing as position vectors (i.e., placing its initial point at the origin), the curve connecting the terminal points of the image vectors is called a space curve, often denoted by C.
The main purpose of this section is to introduce the concept of a vector function and its interpretation as a space curve. You will NOT be asked to draw space curves. However, though, the helix is important (Example 4), and you should be able to recognize a helix from its equation <cos t, sin t, t>. Also, You should be able to recognize a straight line and a line segment .
There will be no exam/hw problems on this section.
Section 13.2. Derivatives of vector functions.
13.2 Derivative Algebraic expression its Geometric meaning
Algebraic calculations for derivatives r'(t) of vector functions are relatively easy.
It is usually easy to calculate r'(t) or T(t) algebraically. You must also know their geometrical meaning, which is that these vectors are tangent to the curve in the direction of the change of r(t), equivalently the direction that the corresponding space curve C moves.
how to find an equation of a line tangent to a space curve given by a vector functions. Key is that the direction vectors can be obtained by either of the tangent vectors and the unit tangent vectors.
See Figure 4 to understand what it means by that r(t) and r'(t) are perpendicular.
Unfortunately, there is no example to practice this important theorem in the textbook. This theorem is crucial to understand the reason why the three vectors T(t), N(t) and B(t) in 13.3 (see below) are perpendicular to each other.
Section 13.3. Part I. T(t), N(t) and B(t), and the Normal & Osculating planes
13.3 TNB and its Geometric meaning
13.3 TNB and its Geometric meaning - full includes the reason why TNB are mutually orthogonal and unit vectors.
Section 13.3. Part II. Arc Length and Arc Length Function will be covered later in the semester.
251PracticeFinal V1 Questions #1, #2. #3 (Not for submission. We will go though the question this coming Wed. 9/18 class when we will do a review session for the Exam 1)
WebAssign: HW 4 Due 9/18 Wed 11:59pm
You will be able to do #1--#10 as of 9/13 Fri.
For #11-#13, you can wait until after 9/16 Mon. class.
Exam 1 on 9/19. See Hour Exam 1
Hour Exam 1
Section 14.1. Functions of several variables.
14.1 Review: Variables and Graphs - full (animated)
Functions in one(1) variable whose graphs are drawn on the 2-d plane; Functions in two(2) variables whose graphs are drawn on the 3-d space; Functions in three(3) variables whose graphs are drawn on the 4-d space
14.1 Level Curves and Level Surfaces - full (animated)
Recall 12.1 12.6 Level Curves and Constructing 3-d Surfaces from Recitation 1.
14.1 Point of view - full (animated)
Any 2-d curve can be viewed as a level curve of z=f(x,y); Any 3-d surface can be viewed as a level surface of w=f(x,y,z)
In Calc III, one of the main purpose is to understand the phenomena in 3-d space. The 2-d level curves is easy to draw and help understanding its 'original' 3-d surface; Interpreting a 3-d surface as a level surface of the 4-d graph will provide a great convenience.
You will not be asked to sketch a 3-d graphs on an exam explicitly. However, you should be able to imagine level curves, level surfaces and abstract objects.
Skip: Cobb-Douglas production function.
Skip: Section 14.2. Limits and continuity.
A limit involves a concept of paths approaching to a target point.
For y=f(x): x is the only variable in the domain. The paths in x-> a have two possible routes; from left and from right
For y=f(x,y): x & y are the two variables in the domain. The paths in (x,y) -> (a,b) has infinitely many possible routes (360 degrees).
Section 14.3. Partial derivatives, Part 1. Algebraic computations
14.3 Partial Derivatives: Definition, Examples, Tree Diagram and 14.3 Partial Derivatives: Definition, Examples, Tree Diagram - full (animated)
I don't expect you to compute the partial derivatives via limits. You can just follow "takeaways" and "tree diagrams".
Notation: 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.
Sections 14.3 & 14.5 Chain Rule
14.3 14.5 Chain Rule and 14.3 14.5 Chain Rule - full (animated)
Always start with a tree diagram.
14.3 14.5 Higher derivatives in Chain Rule and 14.3 14.5 Higher derivatives in Chain Rule - full (animated)
Sections 14.3 & 14.5 Implicit Differentiation ; We started it on 9/23 Mon but couldn't get far. I will finish on 9/25 Wed.
14.1 Exercises #70 (Not available on WebAssign. You can find it in the eBook. )
WebAssign: HW 5 Due 9/25 Wed 11:59pm
Quiz 5 on Thurs 9/26 will be based on the materials that correspond to Homework 5.
Recitation and Quiz 5
Sections 14.3 & 14.5 Implicit Differentiation
Section 14.3 Part 2 Geometric meaning of Partial Derivatives
Section 14.6. The gradient vector and the directional derivatives.
14.6 Formulas and Examples: Typical questions in the gradient vector and directional derivatives and 14.6 Formulas and Examples: Typical questions in the gradient vector and directional derivatives - what's filled out during classes. See below for the details for the formulas and backgrounds.
14.6 Directional Derivatives via Gradient Vector, and Geometric Meaning - full
Remark: I introduced the directional derivatives via the gradient vector but the formal definition of the the directional derivatives is via limits. This formal definition is needed to make sense of its geometric meaning in full.
14.6 Significance of the gradient vector: Maximum Rate of Increase/Decrease - full:
grad f is the vector in the xy-plane if z=f(x,y) (or in the xyz-space if z=f(x,y,z) ) that points in the direction of the maximum rate of increase (or steepest ascent direction) of f and whose magnitude is equal to the rate of increase in this direction.
- grad f (the negative of grad f) is the vector in the xy-plane if z=f(x,y) (or in the xyz-space if z=f(x,y,z) ) that points in the direction of the maximum rate of decrease (or steepest descent direction) of f and whose negative magnitude is equal to the rate of decrease in this direction.
14.6 Significance of the gradient vector: Orthogonality to the level curves/level surfaces - full:
grad f is the vector that is perpendicular to the level curves/level surfaces.
Using this fact, you can find the tangent lines of level curves or tangent planes of level surfaces.
14.6 Velocity vector in Space in the (unit vector) u-direction - animated
The velocity vector in space in the direction of the gradient vector (or any direction) must be adjusted to the unit vector in the same direction.
Skip: Normal lines
14.3 Exercises #75 (Not available on WebAssign. You can find it in the eBook. )
Finish Example 4 and Example 7 in 14.6 Formulas and Examples: Typical questions in the gradient vector and directional derivatives
WebAssign: HW 6 Due 10/2 Wed11:59pm
Quiz 6 on Thurs 10/3 will be based on the materials that correspond to Homework 6.
Recitation and Quiz 6
Section 14.4. 14.4 Differentials
Recap Calculus I on how to find local max/min via Second derivative test
Introduced a similar task that will do in Calculus III: Algorithm to find Local maxima, local minima, or saddle point for two-variable functions f(x,y) using second derivative test.
The definition of D=f_xx f_yy - (f_xy)^2
Local maxima, local minima, or saddle point for two-variable functions f(x,y) using second derivative test.
Section 14.8. Lagrange Multipliers
Optimization problems, i.e., Max/Min. and how to use Lagrange Multiplier Method.
14.8 Lagrange, Case 1 and Case 2: Examples and Steps - blank handout
Wording in Quizzes/Hour Exams (but probably not in the common final)
14.4 Lagrange, Case 1: Ideas, Examples and Figures and 14.4 Lagrange, Case 1: Ideas, Examples and Figures - animated Examples 1 and 2
WebAssign: HW 7 Due 10/9 Wed11:59pm
Exam 2 on 10/10. See Hour Exam 2.
Review session and Exam 2
Section 14.8. Lagrange Multipliers
Sections 15.1--15.2. Double integrals in Rectangular Coordinates
15.1-15.2 Part 1: Evaluating Double Integrals
Iterated integrations
How to express a 2-d region D in inequalities of x and y - full
This is basically the review of Calculus I. A double integral will be evaluated over such D.
Double Integrals to Iterated Integrals depending on the Types of D - full
(Quick) discussion of Riemann sum and Fubini's Theorem - full
We had a brief discussion on Riemann sum though such questions won't be on the exams.
I strongly recommend that, when setting up an iterated integral, always start from the 'raw' integral then move onto an 'iterated' one.
WebAssign: HW 8 Due 10/16 Wed11:59pm
Quiz 7 on Thurs 10/17 will be based on the materials that correspond to Homework 8.
Recitation and Quiz 7
Sections 15.1--15.2. Double integrals in Rectangular Coordinates
15.1-15.2 Part 2: Geometric Applications of Double Integrals
Volume of a solid via Double Integral in Rectangular Coordinates - animated; You should sketch regions and give the relevant inequalities to set up the double integrals.
Handout Ch 15 Surfaces and Solids contains
The surfaces on pp.1-2 that I expect you to remember.
The 3-d solids, with inequalities. on pp.3-4 that appear a lot hence you should feel comfortable with them for the rest of the semester.
The file is meant for Triple Integral toward the end of Ch 15. However, it will still help to understand the volume of solids in double integrals.
Utilize the inequalities for the solids and identify the (1)--(3) as done in class.
(1) under "what function" (2) above "what function" (3) over "what region D"
The area A(D) of a region D by a double integral - animated; The formula (9) in Section 15.2, right above Figure 19 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 Section 15.2, except for formula (9), the area of D using double integral.
Section 15.3. Double integrals in polar coordinates.
Polar areas (from Calc II) and Evaluating Double integrals in Polar coordinates - animated
dA=dx dy or dy dx in Rectangular coordinates; dA=r dr d(theta) in Polar coordinates: Try to understand the reason why dA is equal to dx dy or r dr d(theta) that in different coordinate systems.
Given problem, think over if D is better with rectangular coordinates or polar coordinates
identify items (1)--(3), as done in class;
(1) under "what function" (2) above "what function" (3) over "what region D"
These (1)(2)(3) are translated into three inequalities of x, y, and z, as in the handout.
This homework is not to be submitted.
WebAssign: HW 9 Due 10/23 Wed11:59pm
Quiz 8 on Thurs 10/24 will be based on the materials that correspond to Homework 9.
Note that Homework 9 includes Ch 15 Surfaces and Solids.
Recitation and Quiz 8
Section 15.6 Triple integrals
15.6 Part 1. Evaluating and Setting up Triple Integrals - animated
dV = dx dy dz in rectangular coordinates
Key is how to obtain three inequalities that describe the solid region E; Translate the tasks from the volume in double integrals
I will ask the surfaces and solids within Ch 15 Surfaces and Solids (with variations such as translate or reflection.)
15.6 Part 2. Application: Volume of a 3-d solid by taking the integrand 1 - animated
Skip: The formal definition of triple integrals via limits. Change of the order of triple integrals (Try Example 4 if you wish, which will help you improve visualizing the geometry). Moment of inertia, Density
Sections 15.7 - 15.8, Part I. Cylindrical and Spherical coordinates.
Sections 15.7 - 15.8, Part I. Basics of Cylindrical and Spherical coordinates
In Cylindrical Coordinates, strictly speaking, the variable r takes both positive and negative values as in polar coordinates from Calculus 2. In our course, you will be fine if you consider only r >=0.
Basic "cylindrical" surfaces
Section 15.7: See Figures 4,5,6, and Example 2 (Figure 7)
Basic "spherical" surfaces
Section 15.8: See Figures 2,3,4
Check out the nice animated figure Explore It: A Region in Spherical Coordinates under Resources on WebAssign
Comparing Ch15 Surfaces and Solids commonly appeared in three different coordinate systems; Rectangular, Cylindrical and Spherical.
Sections 15.7 - 15.8, Part II. Triple Integrals in Cylindrical and Spherical coordinates - animated
dV=r z dr d(theta) in Cylindrical coord.
dV= rho^2 sin(phi) d(rho) d(theta) d(phi) in Spherical coord.
Key is to recognize the solid region E in Cylindrical and Spherical coordinates, then obtain three inequalities that describe E; see Ch 15 Surfaces and Solids
I strongly recommend that, when setting up an iterated integral, always start from the 'raw' integral then move onto an 'iterated' one.
Sections 15.7-15.8 Part III. Application: Cylindrical and Spherical Volume 3-d solids - animated
As before, taking the integrand =1.
I strongly recommend that, when setting up an iterated integral, always start from the 'raw' integral then move onto an 'iterated' one.
Obtain the equations of the surface and the inequalities marked as "HW" on the handout.
Obtain, for yourself, the inequalities in three different coordinates that describe E, as done in class.
On the last page of Sections 15.7-15.8 Part III. Application: Cylindrical and Spherical Volume 3-d solids - animated: Five(5) different ways to represent the volume of the solid enclosed determined by the upper hemisphere with radius 1
double integrals in two different coordinate systems (rectangular, polar)
triple integrals in three different coordinate systems (rectangular, cylindrical, spherical)
WebAssign: HW 10 Due 10/30 Wed 11:59pm 11/1 Fri 11:59pm
Quiz 9 on 10/31 Thursday will be on Homework 10
Recitation and Quiz 9
Section 16.2 Line Integrals
16.2 Section and 13.3. Part II. Arc Length - animated
Arc length in a space curve in 3-d space is obtained by a natural generalization from a 2-d curve.
16.2 Line integrals of real valued functions - animated
Line integrals of real valued functions along C w.r.t. arc lengths: Example 1 and Example 5
Line integrals of real valued functions along C w.r.t. x, y (and z): Example 4 and Example 6
Line integrals when C is a piecewise smooth curve: Example 4-variation
16.2 Meaning of Line integrals - full
In the Line Integrals of Vector Fields, note three(3) different formulas; each has its own advantage.
The equalities among these three integrals are given in the handout Formula comparison: Line Integrals vs. Surface Integrals (16.2, 16.6, 16.7)
16.2 Line integrals of vector fields along C / The Work done by force fields - animated
In class, I evaluated only the calculation friendly formula in (g), the first bullet point.
You can try (f) and the second bullet point of (g) via T (unit tangent vector) and confirm the same result.
16.2 Line integral of vector field i s the same as Line integral w.r.t x, y (and z) - full
Skip: Formal definition of line integrals in terms of limits. Mass and the center of mass
Section 16.3 The Fundamental Theorem for Line Integrals (FTLI)
16.3 Conservative vector field - animated
Criteria
How to find f whose gradient vector is equal to the given conservative vector field
16.3 Significance of FTLI - animated
(1) The integral value is independent of the paths but depends only on the initial point and the terminal point of the path.
(2) The integral is zero on a closed curve
eBook 16.3 Exercises 11,12, 13 (I have copied them on the files 16.3 Significance of FTLI - animated and 16.3 How to apply FTLI when calculating line integrals of conservative vector fields; Convenience of FTLI - animated)
WebAssign: HW 11 Due 11/6 Wed 11:59pm 11/8 Fri 11:59pm
No Quiz on 11/7 Thursday. Instead, I will give a regular lecture.
16.6 Parametric Surfaces and their Areas
Parametrizations of surfaces; see odd pages: r is a parametrization from the uv-plane (2-d) to the xyz-space (3-d)
Complete the parametrizations and Partial Derivatives in Parametrizations of surfaces & Partial derivatives of r(u,v)
WebAssign: HW 12 Due 11/13 Wed 11:59pm
Review session (11/13 W) and Exam 3 (11/14 Th)
16.7 Surfaces Integrals, Part 1: Unoriented surface integrals of real-valued functions
16.7 Surfaces Integrals, Part 2: Unoriented surface integrals of vector fields
Handouts (Their filled solutions can be found in the two "animated" lecture notes under 16.7 Part 2 below)
16.7 Concept/Meaning/Tip for Surface intergrals over Oriented surfaces
16.7 Ex4 Flux of VF Oriented Sphere
16.7 Example 5 (Total 8 page file)
16.7 Ex5 Part1 Oriented Paraboloid surface S_1: (a)(b)(c) & (f)(g) on pp.1-3
16.7 Ex5 Part2 Oriented Disk surface S_2: (a)(b)(c) & (f)(g) on pp.4-6
16.7 Ex5 Part3 Oriented Closed Surface surface S=S_1+S_2: (a) & bullet items on pp.7-8
Surface Integrals of Vector Fields over Oriented S - animated
251 Exam 4 Fall23 #4 (not to submit but to practice)
WebAssign: HW 13 Due 11/20 Wed 11/21 Thur11:59pm
16.7 Surfaces Integrals, Part 2: Unoriented surface integrals of vector fields, continued
Section 16.5 Curl and Divergence; Definition & Notations only
Definition of curl and divergence of a vector field: curl(F) is a another vector field and div(F) is a scalar field.
Orientations of a space curve C that is a boundary of a surface S in Stokes Theorem
Example 1
Example 2
Exercise 16.8 #1
Sample Final #11
Section 16.9 Divergence Theorem
Example 1
Example 2
Sample Final #10 (It's similar to eBook Exercise 16.9 #19)
Summary of the Theorems: Moving back & forth between `integral region <-> its boundary` in Ch16
WebAssign: HW 14 Due 11/26 Tue or 11/27 Wed 11:59pm