Two Variable Calculus
Arleen Jerdee
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.[1]
Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.
In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces:
The consequence of the first difference is the difference in the definition of the limit and differentiation. Directional limits and derivatives define the limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators.
The consequence of the second difference is the existence of multiple types of integration, including line integrals, surface integrals and volume integrals. Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot be properly defined.
In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.[2]
Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.
Multivariate calculus is used in the optimal control of continuous time dynamic systems. It is used in regression analysis to derive formulas for estimating relationships among various sets of empirical data.
Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. In economics, for example, consumer choice over a variety of goods, and producer choice over various inputs to use and outputs to produce, are modeled with multivariate calculus.
This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.
The materials have been organized to support independent study. The website includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include:
The best textbook for understanding single variable calculus is subjective and depends on personal learning style and preferences. However, some popular options include "Calculus: Early Transcendentals" by James Stewart, "Single Variable Calculus: Concepts and Contexts" by James Stewart, and "Calculus" by Michael Spivak.
When choosing a textbook, it is important to consider your current level of understanding and the difficulty level of the textbook. Look for textbooks with clear explanations, plenty of examples, and practice problems at varying levels of difficulty to ensure that you can follow along and challenge yourself as needed.
Yes, there are many online resources that can supplement a textbook for single variable calculus. Some popular options include Khan Academy, Paul's Online Math Notes, and MIT OpenCourseWare. These resources offer video lessons, practice problems, and interactive tools to help you better understand the material.
To make the most out of a single variable calculus textbook, it is important to actively engage with the material. This can include taking thorough notes, practicing problems, and seeking help from a tutor or teacher if needed. It is also helpful to regularly review previous material to ensure a strong understanding before moving on to new concepts.
Yes, there are alternative learning methods for understanding single variable calculus. Some options include online courses, study groups, and one-on-one tutoring. It is important to find the method that works best for you and fits your learning style and schedule.
A comprehensive, mathematically rigorous exposition, Single Variable Calculus with Early Transcendentals blends precision and depth with a conversational tone to include the reader in developing the ideas and intuition of calculus. A consistent focus on historical context, theoretical discovery, and extensive exercise sets provides insight into the many applications and inherent beauty of the subject.
Paul Sisson received his bachelor's degree in mathematics and physics from New Mexico Tech and his PhD from the University of South Carolina. Since then, he has taught a wide variety of math and computer science courses, including Intermediate Algebra, College Algebra, Calculus, Topology, Mathematical Art, History of Mathematics, Real Analysis, Mathematica Programming, and Network Operating Systems. He is Professor of Mathematics and Provost Emeritus at Louisiana State University in Shreveport.
Tibor Szarvas received his master's degree in mathematics from the University of Szeged, Hungary, and his PhD from the University of South Carolina. His teaching experience in mathematics includes courses such as Intermediate Algebra, Liberal Arts Mathematics, Mathematics for Elementary Teachers, Precalculus, Calculus, Advanced Calculus, Discrete Mathematics, Differential Equations, College Geometry, Mathematical Modeling, Linear Algebra, Abstract Algebra, Topology, Real Analysis, Complex Variables, Number Theory, and Mathematical Logic. He is currently serving as Professor and Chair of Mathematics at Louisiana State University in Shreveport.
Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances.
Would it be possible to keep up or would i be overwhelmed by a hopeless struggle trying to grasp the new material while at the same time looking back at as well as relearning a few core concepts from single variable calculus? I have rather horrible basic knowledge from single variable, but I do know my way around linear algebra.
The reason for asking is a choice i have between "retaking" single variable this quarter but it collides with the schedule for multivariable calculus. Multivariable is probably much more important for later courses (electromagnetism and field theory) as well as just seem to be a lot more fun and intersting math.
The phrase "multivariable calculus" might be somewhat misleading. This might imply that it is different from single variable calculus, i.e. first you learned SVC, now you can leave that behind and you'll learn MVC. This is not the case. A great example of this is integrating with respect to two variables. Integrating with respect to two variables is a relatively simple and important concept in MVC, but does not rely on some brand new ideas developed only for this field of mathematics that introduce a completely different way of thinking. Integrating with respect to two variables is just integrating with respect to one variable twice. Unsurprisingly, this is no easier than integrating only once.
This brings me to my main point. In MVC, single variable concepts and their applications do not just only up occasionally, they are important in every chapter and every class. They laid the groundwork for all future concepts, and most problems in MVC are solved by reducing them to SVC problems.
In conclusion: As was already mentioned in some of the comments, I don't want to give specific advice for your situation. A friend or teacher who understands you better will be in a better position to give you specific advice. However, I did want to make you aware of exactly what a MVC course is and what will be required of you in terms of single variable knowledge.
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.
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