Calculus 4 Differential Equations
Janita Locklin
Calculus 3 is a branch of calculus that focuses on multivariable calculus, including topics such as partial derivatives, multiple integrals, and vector calculus. Differential Equations, on the other hand, is a branch of mathematics that deals with the study of equations involving derivatives and their applications. While both involve calculus, Calculus 3 is more theoretical and abstract, while Differential Equations is more applied and practical.
It is recommended to take Calculus 3 before Differential Equations, as Calculus 3 builds upon the concepts learned in Calculus 1 and 2, and provides a foundation for understanding Differential Equations. However, some universities may have different course sequences, so it is best to consult with your academic advisor.
While there may be some overlap in topics between Calculus 3 and Differential Equations, they are distinct subjects with different focuses. Calculus 3 mainly deals with multivariable calculus, while Differential Equations focuses on solving equations involving derivatives and their applications.
This may vary for each individual, as some may find Calculus 3 more challenging due to its abstract and theoretical nature, while others may find Differential Equations more challenging due to its application-based approach. It is best to consult with your professor or academic advisor for more specific information.
Both Calculus 3 and Differential Equations have numerous applications in various fields such as engineering, physics, economics, and more. For example, Calculus 3 can be used to analyze and optimize multidimensional systems, while Differential Equations can be used to model and solve problems in areas such as fluid dynamics, population growth, and electrical circuits.
I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for the following reason: I always tell them that they are one of the most important topics for applications of calculus (this is mainly a course for students in the sciences) and that all sorts of fields use them...and then all I have to tell them are things that are to a certain extent quite dull: exponential growth, Newton's law of cooling, the logistic equation, and a few other of the classics. While each of these is quite important and do have broad applications, I've never seen anyone be shocked to learn that populations of rabbits breeding in the wild grow approximately exponentially.
My knowledge of applied fields isn't terrible, but I'm still at a loss as to what plausible models I could teach them about where the global results are not immediately obvious, so I ask: what are some simple differential equations, simple enough for a freshman calculus class, which occur in the sciences and have behavior interesting enough the catch peoples' interest?
People have been shocked precisely that breeding rabbits grow exponentially. In particular the Australians were in the 19th century. In 1859, some guy named Thomas Austin brought 24 rabbits from England so he could amuse himself by shooting at them. In a few years the population had grown to hundreds of millions of rabbits, decimating Australia's native ecology. Oops. (link: _in_Australia)
Another simple example is money. You invest $1,000 in a bank account earning 10% compounded continuously; how much do you have after 30 years? The answer: A whole heck of a lot of money. I think it is not all that intuitive to many people, but the calculus unambiguously proves it.
I have also covered predator-prey models (which are covered in Stewart's Calc I book.) The differential equations are simple to explain, but draw the oval in the phase plane, and to a beginner the behavior is really rather stunning: the populations oscillate! I think the most natural naive guess is that the populations' behavior would settle down to some equilibrium, and this is not what happens. Probably you can even compare the behavior of the ODE's with data from some real world example (I think Stewart has a graph of such data).
Perhaps you're teaching some kind of honors class where this is old hat to the students; I confess that I'm somewhat disagreeing with the premise of your question, so perhaps I've misunderstood the situation you're in. But in my experience, there is a lot of meat in the very simplest examples.
Airy equation! The optical phenomenon of the rainbow was already explained in ancient times by means of reflection and refraction of light within the spherical droplets of rain. But, after Snell's law of refraction, a complete model was available, which gives account of any quantitative aspectof the phenomenon, including the multiple rainbows. Describing the intensity near an optical caustic, led J. B. Airy to the ODE (now called Airy equation or also Stokes equation)$$\ddot u-xu=0$$whose solutions are the special functions Ai(x) and Bi(x). Here's the original Airy's memory on the Transactions of the Cambridge Philosophical Society (Part 3, XVII).
Maybe it would be interesting to discuss partial differential equations that can be reduced to ordinary differential equations, if one uses symmetries. Take the hydrogen atom for example: You get a decomposition in a radial and spherical part, the second can be solved by separation of variables. So you get three ordinary differential equations and their solution will give you a description of the hydrogen atom in non relativistic quantum mechanics.
This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's.
Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress. Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory.
In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182 in this same series, Translations of Mathematical Monographs, and shows the theory "in action".
Graduate students and research mathematicians interested in all areas of mathematics where nonlinear PDE's are used and studied, including algebraic and differential geometry and topology, variational calculus and control theory, mechanics of continua, mathematical and theoretical physics.
This unit opens with topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals, polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, through cylinders, spheres and other parametrised surfaces), Gauss' and Stokes' theorems. The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).
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