III. Vector Calculus
Learning Objectives
- Perform basic vector operations (scalar multiplication, addition, subtraction).
- Express a vector in component form.
- Explain the formula for the magnitude of a vector.
- Describe three-dimensional space mathematically and locate points in space using coordinates.
- Perform vector operations in R3
- Calculate the dot product of two given vectors.
- Find the direction cosines of a given vector.
- Calculate the cross product of two given vectors.
- Find a vector parallel and/or orthogonal to two given vectors.
- Recognize a vector field in a plane or in space.
- Sketch a vector field from a given equation.
- Convert between cylindrical and rectangular coordinates, and between spherical and rectangular coordinates.
- Determine divergence from the formula for a given vector field.
- Determine curl from the formula for a given vector field.
- Use the properties of curl and divergence to determine whether a vector field is conservative.
This chapter was revised and remixed from Calculus Volume 3 by Gilbert Strang and Edwin “Jed” Herman, et al, which is licensed under the CC BY NC SA 4.0 International License.
Vectors are essential tools in mathematics and physics, used to represent quantities that possess both magnitude and direction. This chapter will explore the fundamental concepts of vectors and the operations associated with them, helping you build a foundation for understanding vector calculus and three-dimensional space. By the end of this chapter, you will have the skills to manipulate vectors, represent them in different coordinate systems, and analyze their properties in mathematical and physical contexts.
You will begin by examining basic vector operations, including scalar multiplication, vector addition, and subtraction. We will introduce you to the component form of a vector, a format that allows you to work efficiently with vectors in both two-dimensional and three-dimensional spaces. With these tools, you’ll also learn how to calculate a vector’s magnitude, a measure of its length, and understand the significance of direction cosines, which provide the orientation of vectors in space.
The chapter progresses into three-dimensional space, or R, where we will explore vector operations such as the dot and cross products, which reveal deeper relationships between vectors. The dot product, for instance, is key to determining the angle between two vectors, while the cross product produces a vector orthogonal to both original vectors, providing insight into perpendicular and parallel directions in space.
Beyond basic operations, this chapter will guide you through the concept of vector fields, which represent vector-valued functions at every point in a plane or in space. Understanding vector fields is critical in physics and engineering as they can represent phenomena such as electric and magnetic fields, fluid flow, and force fields. You will learn to recognize, sketch, and interpret vector fields, which is essential for analyzing how forces or flows vary across different regions.
Lastly, we introduce coordinate transformations, including the conversion between cylindrical, rectangular, and spherical coordinate systems. These transformations are valuable tools for visualizing and solving problems involving symmetry in three dimensions. In this context, you will also encounter two important differential operators: divergence and curl. These operations provide valuable insights into the behavior of vector fields. You’ll learn to compute these operators and use them to determine whether a field is conservative, a property that has far-reaching applications in fields such as physics and engineering.
Each of these topics will build upon the last, equipping you with a robust understanding of vectors and vector calculus that is applicable to a wide range of mathematical and physical problems. Prepare to dive into the mathematical description of space and gain tools that will empower your work in calculus, physics, engineering and beyond.
