Day and Date
Monday May 6
Friday May 3
Thursday May 2
Tuesday April 30
Tomorrow we will develop a formula that allows us to easily compute the divergence of a differentiable vector field at any point in its domain.
Monday April 29
Friday April 26
Thursday April 25
Tuesday April 23
Today we reviewed the meaning of the line integral of a vector field along a curve (as developed in the handout on vector curve integrals), took a look at different notations for line integrals, and did one more example. We also noted that we have now studied 6 different types of integrals: single integrals of scalar functions with domains a closed interval, double integrals of scalar functions with domain a "nice" region in the plane, triple integrals of scalar functions with "nice" domains in 3-space, line integrals of scalar functions along a curve (computing totals from density functions), surface integrals of scalar functions with domain "curvy" surfaces in 3-space, and line integrals of vector fields along a curve.
We then took a quick look at an example of the fundamental theorem of line integrals. By noting that if a vector field \( \vec{F} \) is actually the gradient of a scalar field \( \vec{F}=\nabla f \), then the line integral \(\int_C \vec{F} \) depends only on the initial and final points of the curve \(C\) and not on the rest of the curve. So If \(C_1, C_2\) are two different curves with the same initial and final points, then \(\int_{C_1} \vec{F}= \int_{C_2} \vec{F}\). Tomorrow we will state this fundamental theorem carefully and explore some of its implications.
Monday April 22
Today we looked at some images of vector fields: a wind velocity vector field plot and a more sophisticated wind flow plot. We also looked at this handout designed to help us better understand what vector fields are.
We then started discussing the idea of integrating a vector field over a curve. As with other types of integration, we can askToday, we addressed "What is it?" and did one example of How do we compute it? We also spent a brief time on "What does it tell us?. Tomorrow, we'll get more practice in computing and use specific results to get a better feel for what this type of integration tells us. For a preview take a look at this handout on integrating a vector field along a curve. You can wait until after class tomorrow to work on the problems from this handout and the new assignment for problems from Section 14.2.
Friday April 19
In class, we looked at integrating over a surface. A key part of evaluating a surface integral is expressing the area element dA in terms of the coordinates chosen to describe the surface. In the simplest cases, we can use a geometric argument to deduce an expression for dA. Today we also explored a more general procedure --- a computational approach. The basic idea and some examples are on this handout. The handout also has the assigned problems for this material.
As with integrating over a curve, we follow an approach that differs somewhat from the main approach used in the text. The text approaches integrating over curves and surfaces in terms of parametrizing the curve or surface. Our approach of computing dA is a bit more general (and modern). Specifically, we think of gridding our surface by two families of curves where curves from the two families meet non-tangentially and where the differentials \(d\vec{r}=\langle dx, dy \rangle\) for every curve in each family are easy to compute. We then use \(dA =\|d\vec{A}\|= \| d\vec{r_1}\times d\vec{r_2} \| \) as our infinitesimal area element for that surface. You are welcome to read about the text's approach to integrating over a surface in Section 16.4.
Thursday April 18
In class, we looked at integrating over a surface. A key part of evaluating a surface integral is expressing the area element dA in terms of the coordinates chosen to describe the surface. In the simplest cases, we can use a geometric argument to deduce an expression for dA. In other cases, we need a computational approach. The basic idea and some examples are on this handout. The handout also has the assigned problems for this material.
As with integrating over a curve, we will follow an approach that differs somewhat from the main approach used in the text. The text approaches integrating over curves and surfaces in terms of parametrizing the curve or surface. Our approach is a bit more general. You are welcome to read about the text's approach to integrating over a curve in Section 16.2 and integrating over a surface in Section 16.4.
Tuesday April 16
We started class with a first example of integrating over a surface. As part of this, we needed to describe the surface in terms of a chosen coordinate system and then work out an expression for the area element of the surface in that coordinate system. For the first example, the surface was a sphere and we were able to use a geometric argument to deduce an expression for the area element in spherical coordinates. We will develop a more general approach for more general situations. As part of that, we will use a new tool called the cross product of two vectors.
In the latter part of class, we defined the cross product of two vectors. The cross product of \(\vec{u}\) and \(\vec{v}\) is a new vector denoted \(\vec{u}\times\vec{v}\) and defined geometrically in relationship to the parallelogram that has \(\vec{u}\) and \(\vec{v}\) as its edges. Specfically, \(\vec{u}\times\vec{v}\) is defined geometrically by these two properties:
Monday April 15
Friday April 12
Thursday April 11
Tuesday April 9
Monday April 8
Friday April 5
The cylindrical coordinate transformations are:
\[ \begin{eqnarray*} x & = & r \cos \theta \\ y & = & r \sin \theta \\ z & = & z \\ & & \\ x^2 +y^2 & = & r^2 \\ \frac{y}{x} & = & \tan \theta \\ z & = &z \end{eqnarray*} \]We then noted that the graph of \(z=c \) is a plane that is parallel to the x-axis, the graph of \(r=c \) is a right circular cylinder with axis the z-axis and of radius r and the graph of \( \theta =c \) is a vertical plane that starts at the z-axis and extends in the direction of the ray making an angle of \( \theta=c\) with the positive x-axis.
We then looked at spherical coordinates whose defining transformations, where \(\rho \geq 0 \), \( 0 \leq \theta \leq 2 \pi \), and \9 0 \leq \phi \leq \pi\) are:
\[ \begin{eqnarray*} r & = & \rho \sin \phi \\ x & = & \rho \sin \phi \cos \theta \\ y & = & \rho \sin \phi \sin \theta \\ z & = & \rho \cos \phi \\ & & \\ x^2 +y^2+z^2 & =& \rho^2 \end{eqnarray*} \]We also noted that the graph of \( \rho =c \) is a sphere centered at the origin of radius c, the graph of \(\theta = c \) is a vertical half-plane that starts at the z-axis and extends in the direction of the ray making an angle of \( \theta=c\) with the positive x-axis and the graph of \( \phi = c \) is a cone (just half of a double cone) making an angle of \( \phi \) with the positive z-axis.
Exam #3 will be on Thursday April 11. We will use the 80-minute period from 2:00-3:20. The exam objectives are in this handout.
Thursday April 4
Exam #3 will be on Thursday April 11. We will use the 80-minute period from 2:00-3:20.
Tuesday April 2
Exam #3 will be on Thursday April 11. We will use the 80-minute period from 2:00-3:20.
Monday April 1
Today, we discussed the basics of polar coordinates and plotting polar curves. The animation below shows the curve r=cos(2θ) being traced out as θ increases. Notice how the r values are negagive for some angles.
Tomorrow, we'll use our ability to graph polar equations to evaluate
double integrals over regions that are best described in the polar coordinate
system. 
Exam #3 will be on Thursday April 11. We will use the 80-minute period from 2:00-3:20.
Friday March 29
Exam #3 will be on Thursday April 11. We will use the 80-minute period from 2:00-3:20.
Thursday March 28
Today, we looked at examples of triple integrals and the corresponding iterated integrals in three variables. A triple integral involves adding up infinitely many infinitesimal contributions to a total over a solid region of space. To describe this type of region, we need a three-dimensional coordinate system so we end up with an iterated integral in three variables (that is, the three coordinate variables). For the examples we looked at today, we used cartesian coordinates. In some other example, we might find it convenient to use some other coordinate system. This week, we will look at polar coordinates in two dimensions and two other coordinate systems for three dimensions: cylindrical coordinates and spherical coordinates.
Exam #3 will be on Thursday April 11. We will use the 80-minute period from 2:00-3:20.
Tuesday March 26
At the end of the hour we introduced the concept of triple integrals over solid regions in three space. This idea is a natural extension of the use of double integrals and is supported by a version of Fubini's Theorem that is phrased in terms of functions on three variables.
Monday March 25
I also passed out two new Handouts. The Introduction to Density handout gives the basics of what a density function really is and has a few straightforward exercises involving constant density functions. It also has a "reading" exercise designed to help show how often density functions are used even though they might not be referred to as density functions. In this light, if \( f(x,y) \geq 0 \) at every point \( (x,y) \) in the region R, then the double integral you can think of the double integral \[\iint\limits_{R}f(x,y) \ dA \] can be thought of in two different ways depending on the units that are assigned to f(x,y)
The second Handout was on Area Density Problems and provides some example problems whose solution requires that you use a double integral to find the total accumulation of various area density functions over various regions.
Friday March 15
Fubini's Theorem: If \(f(x,y)\) is continuous at all points in a rectangle R of all points (x,y) with \(a \leq x \leq b \mbox{ and } c\leq y\leq d\) except for the points on a finite set of curves of area zero, then \[\iint\limits_{R}f(x,y) \ dA = \int_a^b \left(\int_c^d f(x,y) dy \right) \ dx =\int_c^d \left(\int_a^b f(x,y)\ dx \right) \ dy.\] We then worked two examples.
Thursday March 14
We ended the hour by seeing how we can extend the ideas of Riemann sums to finding the volumes of solids under the graph of a function on two variables \(z=f(x,y) \).
Tuesday March 12
Monday March 11
We then used a sequence of slides to outline a second approach to constrained optimization --- the method of Lagrange multipliers. The underlying intuition of this approach is that a minimizer or maximizer of a constrained problem will occur at a "first" or "last" point where level sets of the objective function meet the constrained domain. We then tracked the gradient fields of the constrained functions level sets and the level sets of the objective function and noted that the minimizers and maximizers occurred at points where these gradients are parallel. Thus, the method for using Lagrange multipliers to solve \[\begin{eqnarray*} \mbox{Maximize: } w & = & f(x,y,z) \\ \mbox{Subj. to: } C & = & g(x,y,z) \end{eqnarray*}\] is to find all points \((x,y,z)\) and all numbers \( \lambda \) that solve the system of equations \[\begin{eqnarray*} \nabla f(x,y,z) & = & \lambda \nabla g(x,y,z) \\ g(x,y,z) & = & C \end{eqnarray*}\]
Friday March 8
We also set up the algebra for showing that a purely quadratic function on two variables \(z=Ax^2 +2Bxy +cy^2 \) can be rewritten in a form recognizable as related to elliptic paraboloids \( z= \frac{1}{A} \left[ (Ax+By)^2 +(AC-B^2)y^2 \right]\). In this form we can see that depending on the signs of \( A\) (or \(C\)) and \( AC-B^2 \), the quadratic is the equation of a paraboloid vertexed at \( (0,0) \) (opening up or down depending on the sign of \(A\) or a parabolic hyperboloid (with saddle point at \((0,0) ) \).
Returning to Taylor polynomials on two variables, we saw that that the second derivative test for functions on two variables comes directly from evaluating the second order Taylor polynomial associated with the function \(f(x,y) \) at a critical point where the gradient is the zero vector. Comparing this polynomial with our rewritten quadratic function on two variables leads directly to the Second Derivative Test for functions on two variables.
We then finished the class period by looking at constrained optimization problems. These are problems where we seek to maximize (or minimize) an objective function \(f(x,y) \) or \(f(x,y,z) \) subject to one or more constraint equations. We worked through most of the first problem on this handout just before the end of class.
Thursday March 7
Tuesday March 5
Monday March 4
Friday March 1
Exam #2 will be on Thursday March 7. I will supply a list of exam objectives either at the end of this week or over the weekend. Here is the Exam #2 objectives handout.
Thursday February 28
We then used a handout to look at how to use differentials to relate (infinitesimally) small changes among variables. Generally, we start with a nonlinear relation among various variables and we then compute a linear relation among the differentials for those variables. Differentials can be thought of as coordinates in the "zoomed-in world". Differentials are always related linearly. Ratios of differentials give rates of change. No limit is needed since the limit process has already been taken care of in "zooming in" process.
For reference, here's an applet that allows you to look at tangent planes for the graph of a function of two variables.
Exam #2 will be on Thursday March 7. I will supply a list of exam objectives either at the end of this week or over the weekend.
Tuesday February 26
We also recalled the fact that differentiability for functions on one variable, \(f(x)\), at a point \(P= x_0\) means that the "zoomed in" portion of the graph of the function near P is indistinguishable from a line -- the tangent line to \(f\) at \(x_0\). The linearization of the function at the point \(x_0\) is the related function \(L(x)= f(x_0)+f'(x_0)(x-x_0)\). We then extended this idea to functions on two variables, \(f(x,y)\), and noted that the two dimensional analog of differentiability is that the "zoomed in" portion of the graph of \(z=f(x,y)\) near a point \((x_0,y_0) \) will be indistinguishable to a plane -- the tangent plane of \(f\) at the point \((x_0,y_0) \). Although the existence of both partial derivatives at a point is not sufficient to guarantee a tangent plane (i.e., differentiability), we noted that a function will be differentiable at \((x_0,y_0) \) if there is an open set containing \((x_0,y_0) \) for which the partial derivatives of \(f\) exist at every point in that open set. In this case, the linearization of the function \(f\) at \((x_0,y_0) \) is the related function \(L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)\). The graph \(z=L(x,y)\) is the tangent plane to \(f\) at \((x_0,y_0)\).
I then passed out a handout designed to help you see that differentiable functions have gradients at every point in their domain and that we can use level sets to draw enough gradients to have a very good understanding of how a function on two variables behaves. Tomorrow, we will focus on differentials and how to use them in a more modern way than is stressed in the textbook.
Here is a link to the results of the midcourse survey. Comments are paraphrased and in boldface font. Obvious results are: five people want more in-class examples; one person does not find the computer generated graphics useful; one person does not find the topics/text/tomorrow or daily notes information useful. Two comments (each from only one person) worth discussion are: have longer due dates for homework and laughter can be a bit disruptive.
Exam #2 will be on Thursday March 7. I will supply a list of exam objectives either at the end of this week or over the weekend.
Monday February 25
We also noted that the gradient operator satisfies the sum, product, and quotient rules that we know work for derivatives on a single variable.
We then reviewed the geometric interpretation and the analytic computation formulas of the directional derivative of a function f at a point P and in the direction of a unit vector u. We finished by noting that the limit definition in the textbook matches our intuition that the ' directional derivative is computing the rate of change of the function along a "slice" of the surface obtained by slicing through P in the direction of u.
Friday February 22
While the reasoning we went through to connect these two things may initially be challenging (especially dealing with the \(\vec{Bob} \) vector), the take-away messages are simple:
Here's a handout outlined the reasoning we discussed in class. A key part of this reasoning is that infinitesimal changes \(df\) in outputs are related to infinitesimal displacements \(d\vec{r}\) by \( df=\vec\nabla f\cdot d\vec{r}\).
This last relation also forms a starting point for thinking about directional derivative. For a function f, we can ask about the rate of change at a particular point in a particular direction. We denote the directional derivative as \(df/ds\) where df represents an infinitesimal rise and ds represents an infinitesimal run. We can compute a directional derivative by finding the component of the gradient vector along the direction of interest. If the unit vector \(\hat{u}\) gives the direction of interest, then the directional derivative is given by \[ \frac{df}{ds}=\vec{\nabla} f\cdot\hat{u}. \] An alternate notation for direction derivative is \(D_{\hat{u}}f\). With this, we can write the result as \[ D_{\hat{u}}f=\vec{\nabla} f\cdot\hat{u}. \] It is important to note that we can think of a directional derivative as the component of the gradient vector in the direction \(\hat{u}\).
Thursday February 21
We then began a discussion about how one might estimate the greatest rate of change of a function on two variables at a particular point. We used carefully labeled slides showing level sets of a temperature function and conjectured that the direction of greatest rate of change of temperature at a point was orthogonal to the level set passing through that point. We also noted that, if a scale for the units of temperature (the outputs of the function) was provided along with a distance scale for the inputs, then we could estimate the magnitude of the greatest rate of change of the function at the given point. Tomorrow we will look at how to derive a formula that we can use to compute this direction and magnitude of the greatest rate of change. I also passed out a handout associated with the slides but one page was missing. Here is a copy of the full handout.
Tuesday February 19
Monday February 18
We then began discussing vector-output functions: that is, functions whose inputs are real numbers but whose outputs are vectors. These functions are also called parametrized curves. We looked at examples that parametrized the unit circle and a circular helix (the slinky). The latter was \[ \vec{r}(t)=\cos(t)\,\hat\imath+ \sin(t)\,\hat\jmath+ t\,\hat k. \]
We then defined the limit of vector-valued functions and noted that \[\lim_{x\rightarrow a} \langle x(t),y(t),z(t) \rangle = \langle \lim_{x\rightarrow a} x(t),\lim_{x\rightarrow a} y(t),\lim_{x\rightarrow a} z(t) \rangle. \] From this, continuity and differentiability of vector-valued functions easily followed. In particular, we can compute derivatives of such functions "by components".
Tomorrow we will look at the physical and geometric meaning of the derivatives of vector-output functions, derive the formula for computing the length of the curve produced by a parametrized path, and introduce one of the most important concepts we will use for the rest of the semester: the length volume element \(d\vec{r}. \)
Friday February 15
Next, we reviewed the component, \(\vec{u} \cdot \vec{e}_{\vec{v}} \) of \( \vec{u} \) in the direction of \(\vec{v}\) and the projection, \( (\vec{u} \cdot \vec{e}_{\vec{v}})\vec{e}_{\vec{v}} \) of the vector \(\vec{u} \) along \(\vec{v} \).
We ended the hour by deriving the "point-normal" form for the equation of a plane \[\vec{n} \cdot \vec{P_0 P}=0\] where \(P_0, P, \vec{n}\) are, respectively, a fixed point on the plane, a variable point on the plane, and a normal vector to the plane. After two examples, we noted that if we have the equation of a plane in standard form \(ax+by+cz+d=0,\) then we can just "read off" the components of a normal vector \(\langle a, b, c \rangle\). Our last example tied together the two topics of the day (projections and point-normal forms for planar equations) by finding the distance between two parallel planes by projecting the vector between a point on one plane and a point on the other onto the common normal vector.
Thursday February 14
Tuesday February 12
As a second example, we developed the vector method of describing the points on a line L. Specifically, If \(P_0 \) is a point on line L and \(\vec{d} \) is a vector in the direction of L, then the position vector \(\vec{OP}\) for an arbitrary point \(P(x,y,z) \) on the line can be written \(\vec{OP}=\vec{OP_0}+\vec{d}\) which has a particularly nice form when written using components (see page 673 of the text).
We then finished the hour by finding a formula for the angle between two geometric vectors based at the same point and using it to define the dot product of two vectors. The geometric and component formulas for the dot product of \( \vec{u} = \langle u_1,u_2,u_3 \rangle\) \mbox{ and } \vec{v}=\langle v_1,v_2,v_3 \rangle \) are \( \vec{u} \cdot \vec{v} =\| \vec{u} \| |\vec{v} \| \cos({\theta})=u_1v_1+u_2v_2+u_3v_3\). Using the first form it is easy to see that two non-zero vectors are orthogonal precisely when their dot product is 0 and that checking whether two vectors are orthogonal is easy if we use the second form.
Monday February 11
We then moved to a more algebraic way of representing vectors by choosing a coordinate system and noting that, if we put the base of an arrow representing a given vector, \(\vec{u} \) at the origin (this arrow is called the position vector), then the tip of the vector specifies a unique point \(P(a,b,c) \). We then introduced the notation \(\vec{u} = \langle a,b,c\rangle \) where \(a,b,c\) are called the components of the vector \vec{u}\).
We ended the hour by defining addition and scalar multiplication of vectors (both are done "by components") and computing some examples.
Friday February 8
Tomorrow we will begin looking at vectors.
Thursday February 7
Tuesday February 5
Monday February 4
Friday February 1
We then reviewed the intuition that the mathematical claim \[\lim_{x\rightarrow a}f(x)=L\] means that, regardless of how x "approaches" a (either from the left or the right), the outputs f(x) are approaching the number L.
In a similar fashion, the intuitive meaning of \[ \lim_{(x,y) \rightarrow (a,b)}f(x)=L\] is: regardless of how (x,y) "approaches" (a,b), the outputs f(x,y) are approaching the number L. The big difference is that there are infinitely many ways for (x,y) to "approach" (a,b). On the other hand, there is a relatively easy way to show that a limit \(\lim_{(x,y) \rightarrow (a,b)}f(x) \) fails to exist. If we can show that there are two different paths of approach to (a,b) that yield different values, then the limit does not exist.
The first exam will be on Thursday February 7. If possible, it will be scheduled for the 80 minute period 2:00-3:20 PM. Please send me an email as to whether or not this works for your schedule as soon as possible.
Thursday January 31
We also talked about using traces (slices of the graph that are parallel to the coordinate planes) to improve graphs of functions on two variables. We then talked about how every z-trace corresponds to a level set in the xy-plane and that these level sets can help us to understand the outputs of the function f even if we don't have a three-dimensional graph.
Tomorrow we will say a bit more about level sets for functions on three variables, introduce terminology describing open and closed sets in higher dimensions and begin talking about limits of multivariable functions.
Click on the following link to see an image giving level curves of temperature over North America. Level curves of temperature
The first exam will be on Thursday February 7. If possible, it will be scheduled for the 80 minute period 2:00-3:20 PM. Please send me an email as to whether or not this works for your schedule as soon as possible.
Tuesday January 29
The first exam will be on Thursday February 7. If possible, it will be scheduled for the 80 minute period 2:00-3:20 PM. Please send me an email as to whether or not this works for your schedule as soon as possible.
Monday January 28
The first exam will be on Thursday February 7. If possible, it will be scheduled for the 80 minute period 2:00-3:20 PM. Please send me an email as to whether or not this works for your schedule as soon as possible.
Friday January 25
In discussing planes today we noted that it is possible to use algebra to determine the equation of a plane if we know three non-collinear points on that plane. This is in sharp contrast to the geometrically motivated methods we developed using the slopes \(m_x^z, m_y^z\).
In class, we also reviewed some basics of ellipses, parabolas, and hyperbolas. In particular, we started from a purely geometric definition for each type of curve and used a coordinate system to get an analytic description. In all three cases, the analytic description is a quadratic equation in two variables. Since they are in the book, we skipped many of the details of how the analytic descriptions follow from the geometric. As a small challenge, you can fill in the steps that we skipped over between the geometric definition and the most common form of an analytic descriptions.
We can also turn this around and ask about the graph of any quadratic equation in two variables. It is a fact that the graph of any quadratic equation in two variables is an ellipse, a parabola, or a hyperbola (or a degenerate case). That is, the graph of any equation of the form \[ Ax^2+2Bxy+Cy^2+Dx+Ey+F=0\qquad A,B,C\textrm{ not all zero} \] is an ellipse, a parabola, or a hyperbola. It can be shown that you can determine which type of curve by computing \(AC-B^2\). If this quantity is positive, the graph is an ellipse. If this quantity is zero and one of D or E is nonzero, the graph is a parabola. If this quantity is negative, the graph is a hyperbola.
Tomorrow, we move to three-dimensions so we will be dealing with quadratic equations in three variables and the corresponding graphs that are surfaces in space.
Thursday January 24
Today we worked on gaining geometric insight into planes and their equations. In particular, in computing, say, \(m_x\), it doesn't matter where we "slice" the plane as long as we hold \(y\) constant.
I will often include mathematics symbols on this page and will be using MathJax to do so. Please let me know if the following looks like the quadratic formula to you. If not, please tell me which browser you are using. \[ x= -b \pm \frac{\sqrt{b^2-4ac}}{2a} \]
Tuesday January 22
You should read the portions of Section 12.2 that deal with spheres and right circular cylinders and start work on the homework on the 3D Basics handout
[Exam 1: Fall 2012___] [Exam 2: Fall 2012___] [Exam 3: Fall 2012___] [Exam 4: Fall 2012___]
[Exam 1: Spring 2008] [Exam 2: Spring 2008] [Exam 3: Spring 2008] [Exam 4: Spring 2008] [Exam 5 with Final: Spring 2008]
