Mathematics 2374 Multi-Variable Calculus

This is where you can read notes from me regarding my lectures.

Day One: Remember to get PlayDough©. The determinant of a matrix is a number.

Day Two: We use determinants to find the cross product of two vectors. The cross product of two vectors is always perpendicular to both vectors and its length equals the area of the parallelogram subtended by the two vectors. The mixed product expressions are equal to each other and give a number which is . . . (You'll find out Wednesday.)

Day Three: . . .in absolute value the volume of the parallelepiped subtended by the three vectors of the mixed product. It can also be calculated by creating a matrix with the vectors as rows or columns and then taking the absolute value of the resulting determinant.

Using projections of vectors we can find the shortest distance from a point to a plane--process over formula.

We see in Euclidean 4-space using different kinds of eyes--eyes of analogy, eyes of perspective and eyes of algebra. Using algebra we saw how to stack 3-spaces in 4-space just like we stack planes in 3-space. Two planes in 4-space can kiss at just one point.

Day Four: Using prespective in 3-space we saw a 4-simplex, the simplest hypersolid in 4-space.

Using planes, cylinders and quadric surfaces we saw real-valued functions of two-variables. These surfaces will be the fodder for the remainder of the semester. They have to become your friends. A surface intersected with a plane parallel to the xy-plane (z = c) traces out a level curve on the surface.

There are also real-valued functions of three-variables. Their partial graphs are solids in 4-space. When a 3-space parallel to our xyz-space (w = c) intersects one of these solids, a level surface is obtained.

Day Five: In the best of all possible worlds surfaces are smooth and so the surfaces have tangent planes that closely approximate them close to the point of tangency. Using partial derivatives we can find the equation of these tangent planes and do approximations if we must. The equation for a tangent plane is: z = f(a,b) + f1(a,b)(x - a) + f2(a,b)(y - b)   where f1 and f2 are the partials with respect to x and y respectively.

The gradient vectors are vectors of partial derivatives of a function.

Day Six: We don't live in the best of all possible worlds. We saw how bad it can get. Limits can be the same number on every linear and quadratic approach to a point, but a cubic approach will give a different answer. Surfaces can be connected together but not smooth. Surfaces can have tangent lines in every direction and still not be smooth. A surface is smooth at a point if it is differentiable at that point. What guarantees this? If the partial derivatives exist and are continuous at that point--this is the best of all possible worlds. Most of the functions we can build from old Calculus I functions have continuous partials. We just have to stay away from the edges of the surfaces and places where we would divide by zero.

Day Seven: For single valued functions of two or three unknowns the gradient can be thought of as the "derivative" particularly when the partials are continuous. For vector valued functions of two or more unknowns the "derivative" is a matrix--really a column vector of gradients (row vectors).

Vector valued functions of a single variable graph as curves where we are pointing from the origin to the points on the curve. The derivative here gives a tangent vector to the curve. From that we can find a vector equation for the tangent line to the curve. We saw the chain rule in its simplest form--the composition of a single valued function of two unknowns with a 2-vector of a single unknown.

Day Eight: If the parameter for a vector valued function is time, then the function can be thought of as a position function where the derivative is now the velocity vector whose magnitude is speed. The second derivative is the acceleration vector.

We generalized the simplest form of the chain rule to handle all meaningful compositions of our functions.

The gradient vector is becoming increasingly important. We have used it to find tangent planes and to calculate derivatives using the chain rule. Now it shows itself in the calculation of directional derivatives. In planes parallel to a unit vector, u , and perpendicular to the xy-plane that intersect a surface defined by z = f(x,y) the directional derivative gives slopes of tangents to the curves of intersection and is calculuate by dotting the gradient vector of f with u.

Also the gradient vector gives the compass directions of maximal ascent on a surface and is perpendicular to level curves on the surface.

Day Nine: Let f be a function of three unknowns, then its "graph" has level surfaces and its gradient will be perpendicular to these level surfaces. This gives us another way to find tangent planes to surfaces.

Once we know how to take partial derivatives of functions, since these are functions, we can take their partials to get higher order partial derivatives. The nice thing is that a function really only has three second order partials because the two mixed partials are equal to each other if the first partials were continuous.

Likewise, we can iterate integrals--integrate with respect to one variable and then integrate that with respect to the other variable. We could have done iterated integrals back in Calculus II if we wanted.

This led us to think about double integrals. A double integral for a function over a rectangular region is the limit of sums of "volumes" of approximating "chimneys" under the surface graph of the function. This is just like a single integral for a function over an interval is the limit of sums of "areas" of appromimating "rectangles" under the curve graph of the function.

If our function is continuous over the rectangle then the double integral can be evaluated with either iterated integral over that rectangle. This is Fubini's Theorem.

Day Ten: What about functions of two unknowns defined over non-rectangular regions--can we integrate them? Yes, right now if the region is y-simple or x-simple or can be decomposed into a finite number of y-simple or x-simple regions. Some regions are both y-simple and x-simple, so we can integrate either way to get the answer. We practiced several of these.

The two most important properties of double integrals are that if the function being integrated is non-negative over the region, the double integrals gives the volume trapped below the surface graph of the function over the region in question and the double integral of 1 gives the area of the region.

Day Eleven: We practiced on one more double integral where the region was both y-simple and x-simple. We reviewed the Mean Value Theorem and the Average Value Theorem from Calculus I and then saw the Average Value Theorem for Double Integrals. And then we proved the simple form of our Chain Rule using the Mean Value Theorem from Calculus I.

Day Twelve: Triple Integrals are just like double integrals just one dimension up. We can't see the geometry in total anymore, but the triple integral is still the limit of sums of function values of three unknowns multiplied by a change in volume. So if our function is f(x,y,z) = 1 then its triple integral over a solid will give the volume of that solid.

We have basically three types of solids depending on whether z or x or y is the dependent variable when viewing the solid. We met the Plumber's Delight, My Solid and The Napkin Ring.

Day Thirteen: We were able to find the volume of a piece of Gouda cheese six different ways. And then we found out that although the piece of the napkin ring in the first octant is a type one solid its shadow in the xy-plane is neither y-simple nor x-simple.

We reviewed curves in 3-space, developed the arclength integral, and if our parameter is time, we had distance travelled. Indeed, the derivative of total distance travelled is speed!

Day Fourteen: You saw me, the mathematician, do two motion problems which you probably already knew how to do.

We reviewed the kinds of functions we have seen and when we don't draw vectors in standard position. This led to a new kind of function: Vector Fields where points go in but vectors come out--it is the mathematical way of seeing force fields.

The "del" operator is a vector of partial derivative symbols--they are like verbs without a subject. The gradient of a function of three unknowns is like scalar multiplication. We then define "del dot a Vector Field" and "del cross a Vector Field." The first gave us the divergence of the field; the second gave us the curl of the field. The divergence is a scalar in terms of x, y and z; the curl is a vector in terms of x, y and z.

Day Fifteen: We met "line" integrals of the first and second kind. Those of the first are curves dipped in real-valued functions; those of the second are curves dipped in vector-fields. If we integrate the function f(x,y,z) = 1 over a curve, we find the curve's arc length; if we integrate density over a curve, we find the mass of the curve; if we integrate height, h(x,y), of a surface over a curve in the xy-plane, we find the area of the surface. If we think of a vector field as a force field and if we integrate that force field over the curve, we calculate work done moving along the curve in the force field.

We saw some properties of line integrals, the greatest of which was The Fundamental Theorem of Line Integrals.

Day Sixteen: There was a little review of line integrals today, but the major portion was Green's Theorem. This great theorem equates the line integral around the boundary of a region and a double integral over the region.

The line integral of (1/2)(-y dx + x dy) around the boundary of a region will give you the area of the region.

SPRING BREAK

Day Seventeen: How to use Green's Theorem: Sometimes if the region is nice but its boundary is not so nice, the double integral is better than the line integral. Sometimes the integrand of the double integral is not nice, so the line integral might be better. And we can use just the right line integral to find the area of the region. I then proved one half of Green's Theorem leaving the other half for you.

A conservative vector field has the property that any line integral between two points always gives the same number--i.e. the line integral is path independent. Every conservative vector field is a gradient field. Every gradient field is a conservative vector field. It is easy to check to see if a planar vector field is conversative--the integrand for Green's Theorem must be 0.

Day Eighteen: It is harder to check a 3-D vector field for path independence. We have to take the anti-partial derivatives of P wrt x, of Q wrt y, and of R wrt z and then create from these a function that <P, Q, R> because its gradient field.

We saw the cylindrical co-ordinate system where the grid is made of cylinders around the z-axis, planes passing through the z-axis perdendicular to the xy-plane, and planes parallel to the xy-plane. We saw the cylindrical equations for spheres, cones and circular paraboloids.

Finally, we say the spherical co-ordinate system where the grid is made of spheres centered on the origin, half-planes through the z-axis perdendicular to the xy-plane, and half-cones centered on the origin. Remember the important triangle that inter-related rho, r, z and phi. We saw the spherical equations for spheres, cones and circular paraboloids. Good luck on the test!

Days Nineteen and Twenty: The Change of Variable Theorem for Double and Triple Integrals was the theme of the week. One is given an integral over something "ugly." We use a transformation to pull it back to something "nice" and then integrate the function of the transform times the absolute value of the Jacobian of the transformation over that "nice" thing. We did five examples of double integrals and three examples of triple integrals. The polar and cylindrical Jacobian is   r   ; the spherical Jacobian is   rho squared time the sine of phi  .

Day Twenty-one: We learned how to parameterize surfaces in 3-space. They are ribbons in space. The input is a set of two parameters, u and v; the output is a vector. We are pointing to the points on the ribbon. If one parameter is held fixed we get a curve in space; as it varies we get different curves all "shaped" the same; we can think of the ribbons as these curves glued together. Reversing the parameters gives as another family of curves of similar "shape;" we can also think of the ribbons as these curves glued together.

We saw many examples: some familar surfaces and some new surfaces. We also saw how to parameterize a surface of revolution.

Day Twenty-two: First we saw how to parametrize a torus (bagel, doughnut).

We then developed the double integrals to find surface area. The first was for cartesian functions; the second was for parametrized surfaces.

So, we assign a number to each point of a surface, say with the function, w = f(x,y,z). We can then accumulate this function over our surfaces. This is a surface integral of the first kind. We learned how to convert this to a double integral. If the function is w = 1, the surface integrals gives surface area. If the function is density, the surface integrals gives us the mass of the surface.We did some examples.

Day Twenty-three: Let's suppose we have a surface in the xy-plane and let's also suppose that we have a function that gives the height of a wall over that surface, then the surface integral of the height function over the surface gives the volume of the wall.

We then generalized surfaces integrals to vector fields. They followed the same way as line integrals.

This is called a "flux" integral. I like to call them surface integrals of the 2nd kind. We did several examples.

Day Twenty-four: If we have a surface with a boundary in 3-space, we could do a line integral on the boundary or a surface integral over the surface. Are they related? Yes, by Stokes' Theorem. The line integral of a vector field over the boundary is equal the surface integral of the curl of that vector field over the surface. The vector field has to be continuously differentiable and the surface has to be oriented properly.

We met the Mobius strip that cannot be oriented properly.

We did two examples of Stokes' Theorem.

Day Twenty-five: We did the quarter ramp problem checking both sides of the equation in Stokes' Theorem. We were lucky that three of the line integrals were 0.

We saw where curl got its name and a quasi-proof of Stokes' Theorem.

We were introduced to Gauss's Theorem and we checked it on an inverted cone.

Day Twenty-six: We checked Gauss's Theorem on a closed cylinder--person-on-the-street kind: one triple integral equaled the sum of three surfaces integrals.

Again we saw a quasi-proof of Gauss's Theorem.

If we integrate the force field (1/3)< x,  y,  z >  over the surface of a solid, we will find the volume of that solid. We found the volume of a sphere this way, and we found the volume of a cone of radius 1 and height 1 this way.

The next test is this coming Wednesday.

Day Twenty-seven: We reviewed Taylor series, Taylor polynomials and Taylor's Theorem. We then found the first and second degree Taylor polynomials in two unknowns. First degrees were our old friends tangent planes; second degrees are tangent paraboloids. The Discriminant tells us whether they are elliptic paraoloids or hyperbolic paraboloids. We can use them to find quadratic approximations for the original function near the point of tangency.

Day Twenty-eight: We first did an example of a quadratic approximation.

Critical points of a function are points where both partial derivatives are zero.

The Second Derivative Test allows us to test these points for low points, high points and saddle points on the surface graph of the function. At low points the z values is a relative minimum of the function. At high points the z values is a relative maximum of the function. At saddle points in one direction the z value is minimal but in the other direction it is maximal.

We did two examples of regular functions and then started doing optimization problems. The first was finding the closest points between two lines in three space.

Day Twenty-nine: More optimization problems.

Day Thirty: Review for the final!