A Complete Guide to Integral Calculus (Advanced Calculus)

Mastering Integral Calculus: Jacobians, Gamma Functions, and Surface Integrals

What you'll learn

• Understand and apply the concept of Jacobians in polar coordinates to transform functions of two variables.
• Evaluate double integrals by changing variables, simplifying computations for complex regions.
• Apply double integrals to solve real-world problems, including area, mass, and center of mass calculations in polar coordinates.
• Analyze and evaluate integrals related to the Gamma function and its properties.
• Comprehend the properties and applications of the Gamma function and use it in integral calculus problems.
• Apply the Laplace transform to solve integrals and understand its importance in mathematical modeling.
• Understand Jacobians in cylindrical and spherical coordinates and apply them for coordinate transformations in three-dimensional problems.
• Perform triple integrals by changing variables to cylindrical or spherical coordinates to simplify integrals.
• Apply triple integrals to real-world contexts, such as calculating volumes, masses, and centers of mass for 3D objects in cylindrical or spherical coordinates.
• Calculate surface area using integration techniques for surfaces defined parametrically or in coordinate systems.
• Evaluate surface integrals in cartesian, cylindrical, and spherical coordinates.
• Apply surface integrals to solve problems in physics and engineering, such as flux and surface area calculations in various coordinate systems.

Requirements

• Proficiency in Single-Variable Calculus (Calculus 1 and 2): A solid understanding of differentiation and integration for single-variable functions, including the Fundamental Theorem of Calculus, techniques of integration, and applications of single-variable integrals.
• Introductory Multivariable Calculus Knowledge (Calculus 3): Familiarity with partial derivatives, double and triple integrals, and basic coordinate transformations (e.g., Cartesian to polar).
• Basic Linear Algebra Skills: Understanding of vectors and matrices, which is helpful for transformations and working with Jacobians.

Description

How This Course Works

This course, A Complete Guide to Integral Calculus (Advanced Calculus), builds upon foundational calculus to dive deeply into the behavior and applications of functions in multiple dimensions. Focusing on integral calculus, it also provides an introduction to key vector calculus concepts that follow in the complete series, including major theorems like Green’s, Stokes’, and the Divergence Theorem. This course is essential for students in mathematics, physics, engineering, and other fields who want a practical and theoretical understanding of multivariable calculus tools for real-world problem-solving.

Who Should Take This Course?

This course is ideal for university students currently enrolled in Advanced Calculus, or those who have completed Calculus III and Linear Algebra. It’s also designed for anyone eager to explore multivariable calculus applications more deeply, especially in fields where calculus techniques are indispensable.

Course Overview

This course includes Integral Calculus, the first part of the complete Advanced Calculus series. You’ll have access to lecture videos, whiteboard notes, and problem sets with solutions. Topics covered here include:

Integral Calculus (Sections Included in This Course)

• Two-Variable Functions: Explore Jacobians in polar coordinates, variable changes in double integrals, and applications of double integrals.
• Gamma Function and Laplace Transform: Understand a key integral related to the Gamma function, the Gamma function itself, and the Laplace transform.
• Three-Variable Functions: Learn about Jacobians in cylindrical and spherical coordinates, transformations in triple integrals, and applications of triple integrals.
• Surface Area and Surface Integrals: Calculate surface areas and evaluate surface integrals in Cartesian, cylindrical, and spherical coordinates.

Vector Calculus, Integral Theorems, and Differential Calculus of Several Variables (Available in the Complete Course)

• Vector and Scalar Fields
• Line Integrals
• Flux, Circulation, and Vector Operators
• Integral Theorems
• Introduction to Partial Differential Equations
• Topics in Differential Calculus of Several Variables

Course Content

• Videos: Each topic is introduced and explained thoroughly, with step-by-step examples, making complex problems manageable.
• Notes: Downloadable lecture notes for each section to allow for offline review and reinforcement.
• Assignments: Practice problem sets with solutions so you can check your work after attempting each problem yourself.

Highlights of What’s Included

• Downloadable lecture videos and notes for anytime review.
• Two comprehensive problem sets with solutions to reinforce learning.
• An instructor committed to your success every step of the way.

See you inside the course!

– Gina Chou :)

Who this course is for:

• Undergraduate and graduate students in mathematics, physics, engineering, and related fields.
• Seek a deeper understanding of integral calculus concepts and techniques for multi-variable functions.
• Wish to develop skills in using advanced calculus tools like Jacobians, Gamma functions, and Laplace transforms to simplify and solve complex integrals.
• Aim to apply integral calculus to real-world problems in areas such as fluid dynamics, electromagnetism, and probability theory.