Calculus is a subject that many students often struggle with. Unless you take AP calculus in high school, your first encounter with calculus will probably be in your first year. Even students who have done well in high school math often find calculus notoriously tricky. This is because calculus introduces a part of mathematics that was previously completely foreign to you. With the other courses you take with calculus, coping with the course material can be quite a challenge.
In this article, we will pick the final installation in your calculus course, calculus III, and take it apart, discussing what you should expect. You also learn how you should study and how difficult it is precisely, answering most of the commonly asked questions about the course.
What are the Topics in Calculus 3?
What are the Topics in Calculus 3?Calculus is usually broken down into three parts, delivered over three semesters, with each piece tackling more advanced concepts than the one preceding it. Each part bombards students with a barrage of theorems and formulas. That spans from limits in Calculus I to triple integrals in Calculus III, making calculus courses a daunting task for almost everyone.
Although there might be a difference in one or two concepts being differently distributed, generally, the topics in the calculus courses are split as follows:
Calculus I:
- Limits
- Derivatives and their applications
- Introduction to anti-derivatives
- L’Hopital’s Rule
Calculus II:
- Polynomials
- Integration techniques
- Trigonometric, exponential, and logarithmic functions
- Polar coordinates
- Infinite sequences and series
- Linear equations
- Ordinary differential equations
Calculus III:
- Multivariable calculus
- Partial derivatives
- Vector and Scalar fields
- Higher order derivatives
- Maclaurin and Taylor series
- Differential equations
Notice that each subsequent installation of the course builds upon concepts learned in the previous, adding on additional layers of complexity.
How Difficult is Calculus 3?
So how hard is Calculus 3? Let’s look at the Calculus 3 syllabus and break it down. Calculus 3 mainly deals with multivariable calculus. This means instead of functions with a single input, you now deal with processes whose output depends on multiple variables.
This adds a layer of complexity, mainly when dealing with integrals and functions’ derivatives. The previously learned techniques in Calculus I and II need to be modified to accommodate the additional variables. Moreover, your first encounter with vector and scalar fields will probably be in Calculus 3. One common problem students have in Calculus III is visualizing problems in 3 dimensions.
Which is Harder: Calculus II or Calculus III?
We can start comparing now that we are familiar with what each course brings. The previous question is somewhat controversial, with half the students claiming calculus II is more complicated than Calculus III. At the same time, the other half argues the contrary.
In this Quora thread, we can see both sides have valid arguments. Some say Calculus II is the more challenging course due to the sheer volume of new concepts being introduced. Students who digest new ideas and techniques and apply them often find Calculus II difficult.
Conversely, the other side argues that Calculus III is unequivocally the more challenging course due to the complexity of the subject matter. Previously, all the problems generally contained one variable. Calculus 3 topics involve the manipulation of multiple variables, vector and scalar fields, and partial derivatives.
Students with trouble visualizing problems in three dimensions often find Calculus III a more daunting prospect than Calculus II. Is calculus 3 harder than 2? Or is Calculus 2 harder than 3? At the end of the day, it depends on who is studying it.
What Makes Calculus III the Hardest Class?
This question is for those who are troubled by the extra dimensions. What makes Calculus 3 the most demanding class for some students is obvious: the different dimensions.
Students who find Calculus 2 more difficult often find it the more challenging course because it requires a fair bit of complex algebraic manipulations. Conversely, students who find Calculus 3 the most complex class have trouble visualizing multiple dimensions and complex 3-dimensional shapes.
But many wonder, is Calculus 3 harder than linear algebra? It all is subjective, as each person has his own strengths and weaknesses. However, calculus 3 is generally regarded as a more difficult course. However, research shows that more students fail college algebra as compared to those failing Calculus 1.
How to Study Calculus 3: 7 Tips
Are you worried your GPA might suffer because of calculus 3? Is the prospect of multivariable calculus downright daunting? Don’t worry. We have combined tips from our calculus experts on how exactly you can weather the calculus 3 storm and how you can do quite well. Without further ado, let us dive in:
1. Establishing a solid foundation
Yes, this tip seems a bit cliche, but we can’t understate the importance of building a solid base. To properly understand and excel in multivariate calculus, it is essential to ensure your previous concepts are vital. These include a firm grasp of trigonometry, geometry, and algebra fundamentals, as well as the images learned in the last calculus classes.
2. Good understanding of the previous courses
Building upon the previous point, calculus III builds upon a foundation previously laid by calculus I and II. You should be good with basic calculus concepts and theorems. Furthermore, being comfortable with integration techniques learned in calculus II is paramount to your success with double and triple integrals. You often break apart and solve that by employing previously known methods.
3. Learn how to effectively set up problems
Learning how to solve a problem most efficiently is essential to success in this course. Before solving a problem, you should first take a minute to set it up. For example, if it is a triple integral, the first step would be to sketch it out.
Sketch a graph and find the region over which you are integrating. Decide which coordinate system would make your job the easiest. A notoriously tricky problem in cartesian coordinates might be pretty simple if you switch to spherical coordinates.
Therefore, do not dive in immediately and try to brute force your way through a problem. Instead, please take a minute to analyze it, set it up, and then adopt an approach that makes the problem at hand the simplest.
4. Practice, practice, practice
Again, this might seem cliche, but the only way to effectively grasp and become comfortable with complex concepts is to practice them as much as possible. Practicing problems also helps you identify fallacies in your understanding. The more you practice, the more comfortable you will be with the subject matter and, consequently, the better you will do in the course.
5. Analyze every mistake
Building upon the previous point, it is crucial to analyze your mistakes when practicing. Understanding where you went wrong in a problem will help you identify a weak point and ensure you don’t make the same mistake again on a quiz or an exam.
Make it a goal to not look up the solutions to a problem unless you have spent at least 15-20 minutes solving it. This will ensure you develop your problem-solving skills and not become dependent on external help each time you get stuck.
6. Adopt a proactive approach
It always helps to adopt a proactive approach to learning. Going over the concepts before a lecture and noting points of confusion and ambiguity to clarify in class can be immensely helpful. If the course does not explain your concerns, review them with your instructor.
Developing crystal clear concepts in class will reduce the time you will have to spend going over your notes and books and understanding them later, which you can devote to practice instead.
7. “Two heads are better than one”
Finally, forming a study group with your classmates can significantly help. Creating a group can help you compare notes with classmates, helping clarify essential concepts. Moreover, each individual has particular strengths and weaknesses. Chances are, a friend of yours has a good understanding of a concept you struggle with, and you have a good account of a concept they work with.
Each group member brings their unique thinking to the table and may raise questions you never considered. Finally, studying in a group can help you avoid procrastination, as group members can hold each other accountable.
Wrapping Things Up: How Hard is Calculus 3?
To conclude, yes, calculus 3 can be a daunting course. The same holds true for any new thing you are learning, however. Just like learning a new skill, with determination and consistent hard work, you can master the concepts and ace the course.
After going through it, you will realize you look at the world differently. Calculus 3 introduces you to a part of mathematics prevalent in every field, and going through it can be a gratifying experience when done right. So to conclude, yes. You might find Calculus 3 difficult. But with hard work and our expert tips, you can get through it with another “A” on your college transcript.
Preparing for AP calculus exam? We’ve created a helpful guide to help you out.
If you found this helpful, check out our other high school study tips here.
