Understanding Multivariable Calculus: Problems, Solutions, and Tips
Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogota, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.
01: A Visual Introduction to 3-D Calculus
Review key concepts from basic calculus, then immediately jump into three dimensions with a brief overview of what you'll be learning. Apply distance and midpoint formulas to three-dimensional objects in your very first of many extrapolations from two-dimensional to multidimensional calculus, and observe some of the new curiosities unique to functions of more than one variable.
02: Functions of Several Variables
What makes a function "multivariable?" Begin with definitions, and then see how these new functions behave as you apply familiar concepts of minimum and maximum values. Use graphics and other tools to observe their interactions with the xy-plane, and discover how simple functions such as y=x are interpreted differently in three-dimensional space.
03: Limits, Continuity, and Partial Derivatives
Apply fundamental definitions of calculus to multivariable functions, starting with their limits. See how these limits become complicated as you approach them, no longer just from the left or right, but from any direction and along any path. Use this to derive the definition of a versatile new tool: the partial derivative.
04: Partial Derivatives-One Variable at a Time
Deep in the realm of partial derivatives, you'll discover the new dimensions of second partial derivatives: differentiate either twice with respect to x or y, or with respect once each to x and y. Consider Laplace's equation to see what makes a function "harmonic."
05: Total Differentials and Chain Rules
Complete your introduction to partial derivatives as you combine the differential and chain rule from elementary calculus and learn how to generalize them to functions of more than one variable. See how the so-called total differential can be used to approximate ?z over small intervals without calculating the exact values.
06: Extrema of Functions of Two Variables
The ability to find extreme values for optimization is one of the most powerful consequences of differentiation. Begin by defining the Extreme Value theorem for multivariable functions and use it to identify relative extrema using a "second partials test"-which you may recognize as a logical extension of the "second derivative test" used in Calculus I.
07: Applications to Optimization Problems
Continue the exploration of multivariable optimization by using the Extreme Value theorem on closed and bounded regions. Find absolute minimum and maximum values across bounded regions of a function, and apply these concepts to a real-world problem: attempting to minimize the cost of a water line's construction.
08: Linear Models and Least Squares Regression
Apply techniques of optimization to curve-fitting as you explore an essential statistical tool yielded by multivariable calculus. Begin with the Least Squares Regression Line that yields the best fit to a set of points. Then, apply it to a real-life problem by using regression to approximate the annual change of a man's systolic blood pressure.
09: Vectors and the Dot Product in Space
Begin your study of vectors in three-dimensional space as you extrapolate vector notation and formulas for magnitude from the familiar equations for two dimensions. Then, equip yourself with an essential new means of notation as you learn to derive the parametric equations of a line parallel to a direction vector.
10: The Cross Product of Two Vectors in Space
Take the cross product of two vectors by finding the determinant of a 3x3 matrix, yielding a third vector perpendicular to both. Explore the properties of this new vector using intuitive geometric examples. Then, combine it with the dot product from Lecture 9 to define the triple scalar product, and use it to evaluate the volume of a parallelepiped.
11: Lines and Planes in Space
Turn fully to lines and entire planes in three-dimensional space. Begin by defining a plane using the tools you've acquired so far, then learn about projections of one vector onto another. Find the angle between two planes, then use vector projections to find the distance between a point and a plane.
12: Curved Surfaces in Space
Beginning with the equation of a sphere, apply what you've learned to curved surfaces by generating cylinders, ellipsoids, and other so-called quadric surfaces. Discover the recognizable parabolas and other 2D shapes that lay hidden in new vector equations, and observe surfaces of revolution in three-dimensional space.
13: Vector-Valued Functions in Space
Consolidate your mastery of space by defining vector-valued functions and their derivatives, along with various formulas relating to arc length. Immediately apply these definitions to position, velocity, and acceleration vectors, and differentiate them using a surprisingly simple method that makes vectors one of the most formidable tools in multivariable calculus.
14: Kepler's Laws-The Calculus of Orbits
Blast off into orbit to examine Johannes Kepler's laws of planetary motion. Then apply vector-valued functions to Newton's second law of motion and his law of gravitation, and see how Newton was able to take laws Kepler had derived from observation and prove them using calculus.
15: Directional Derivatives and Gradients
Continue to build on your knowledge of multivariable differentiation with gradient vectors and use them to determine directional derivatives. Discover a unique property of the gradient vector and its relationships with level curves and surfaces that will make it indispensable in evaluating relationships between surfaces in upcoming lectures.
16: Tangent Planes and Normal Vectors to a Surface
Utilize the gradient to find normal vectors to a surface, and see how these vectors interplay with standard functions to determine the tangent plane to a surface at a given point. Start with tangent planes to level surfaces, and see how your result compares with the error formula from the total differential.
17: Lagrange Multipliers-Constrained Optimization
It's the ultimate tool yielded by multivariable differentiation: the method of Lagrange multipliers. Use this intuitive theorem and some simple algebra to optimize functions subject not just to boundaries, but to constraints given by multivariable functions. Apply this tool to a real-world cost-optimization example of constructing a box.
18: Applications of Lagrange Multipliers
How useful is the Lagrange multiplier method in elementary problems? Observe the beautiful simplicity of Lagrange multipliers firsthand as you reexamine an optimization problem from Lecture 7 using this new tool. Next, explore one of the many uses of constrained optimization in the world of physics by deriving Snell's Law of Refraction.
19: Iterated integrals and Area in the Plane
With your toolset of multivariable differentiation finally complete, it's time to explore the other side of calculus in three dimensions: integration. Start off with iterated integrals, an intuitive and simple approach that merely adds an extra step and a slight twist to one-dimensional integration.
20: Double Integrals and Volume
In taking the next step in learning to integrate multivariable functions, you'll find that the double integral has many of the same properties as its one-dimensional counterpart. Evaluate these integrals over a region R bounded by variable constraints, and extrapolate the single variable formula for the average value of a function to multiple variables.
21: Double Integrals in Polar Coordinates
Explore integration from a whole new perspective, first by transforming Cartesian functions f(x.y) into polar coordinates defined by r and ?. After getting familiar with surfaces defined by this new coordinate system, see how these coordinates can be used to derive simple and elegant solutions from integrals whose solutions in Cartesian coordinates may be arduous to derive.
22: Centers of Mass for Variable Density
With these new methods of evaluating integrals over a region, we can apply these concepts to the realm of physics. Continuing from the previous lecture, learn the formulas for mass and moments of mass for a planar lamina of variable density, and find the center of mass for these regions.
23: Surface Area of a Solid
Bring another fundamental idea of calculus into three dimensions by expanding arc lengths into surface areas. Begin by reviewing arc length and surfaces of revolution, and then conclude with the formulas for surface area and the differential of surface area over a region.
24: Triple Integrals and Applications
Apply your skills in evaluating double integrals to take the next step: triple integrals, which can be used to find the volume of a solid in space. Next, extrapolate the density of planar lamina to volumes defined by triple integrals, evaluating density in its more familiar form of mass per unit of volume.
25: Triple Integrals in Cylindrical Coordinates
Just as you applied polar coordinates to double integrals, you can now explore their immediate extension into volumes with cylindrical coordinates-moving from a surface defined by (r,?) to a cylindrical volume with an extra parameter defined by (r,?,z). Use these conversions to simplify problems.
26: Triple Integrals in Spherical Coordinates
Similar to the shift from rectangular coordinates to cylindrical coordinates, you will now see how spherical coordinates often yield more useful information in a more concise format than other coordinate systems-and are essential in evaluating triple integrals over a spherical surface.
27: Vector Fields-Velocity, Gravity, Electricity
In your introduction to vector fields, you will learn how these creations are essential in describing gravitational and electric fields. Learn the definition of a conservative vector field using the now-familiar gradient vector, and calculate the potential of a conservative vector field on a plane.
28: Curl, Divergence, Line Integrals
Use the gradient vector to find the curl and divergence of a field-curious properties that describe the rotation and movement of a particle in these fields. Then explore a new, exotic type of integral, the line integral, used to evaluate a density function over a curved path.
29: More Line Integrals and Work by a Force Field
One of the most important applications of the line integral is its ability to calculate work done on an object as it moves along a path in a force field. Learn how vector fields make the orientation of a path significant.
30: Fundamental Theorem of Line Integrals
Generalize the fundamental theorem of calculus as you explore the key properties of curves in space as they weave through vector fields in three dimensions. Then find out what makes a curve smooth, piecewise-smooth, simple, and closed. Next, manipulate curves to reveal new, simpler methods of evaluating some line integrals.
31: Green's Theorem-Boundaries and Regions
Using one of the most important theorems in multivariable calculus, observe how a line integral can be equivalent to an often more-workable area integral. From this, you will then see why the line integral around a closed curve is equal to zero in a conservative vector field.
32: Applications of Green's Theorem
With the full power of Green's theorem at your disposal, transform difficult line integrals quickly and efficiently into more approachable double integrals. Then, learn an alternative form of Green's theorem that generalizes to some important upcoming theorems.
33: Parametric Surfaces in Space
Extend your understanding of surfaces by defining them in terms of parametric equations. Learn to graph parametric surfaces and to calculate surface area.
34: Surface Integrals and Flux Integrals
Discover a key new integral, the surface integral, and a special case known as the flux integral. Evaluate the surface integral as a double integral and continue your study of fluid mechanics by utilizing this integral to evaluate flux in a vector field.
35: Divergence Theorem-Boundaries and Solids
Another hallmark of multivariable calculus, the Divergence theorem, combines flux and triple integrals, just as Green's theorem combines line and double integrals. Discover the divergence of a fluid, and call upon the gradient vector to define how a surface integral over a boundary can give the volume of a solid.
36: Stokes's Theorem and Maxwell's Equations
Complete your journey by developing Stokes's theorem, the third capstone relationship between the new integrals of multivariable calculus, seeing how a line integral equates to a surface integral. Conclude with connections to Maxwell's famous equations for electric and magnetic fields-a set of equations that gave birth to the entire field of classical electrodynamics.