The calculus of functions \(f:\R \to \R\) like \(f(x)=x^3+3x^2-5x+2\) is the topic of Calculus 1. These functions map (1D) points to (1D) points. Chapter 1 presented the vectors and points and how they are related. The calculus of functions \(P:\R \to \R^n\) like \(P(t)=(t^2,t^3)\) followed. These functions map a parameter (not graphed) to points or vectors (n-dimensional). Chapter 2 to Chapter 4 presented the calculus of functions \(g:\R^n \to \R\) like \(g(x,y,z)=x^3+y^3+z^3+xyz\text{.}\) These functions map points (2D, 3D, or higher) to points (1D). This chapter presents the calculus of functions \(\vec{v}:\R^n \to \R^m\text{.}\) These functions map points (n-dimensional) to vectors (m-dimensional). Because the outputs represent vectors, integration is significantly different.
Subsection5.1.1Presentation
Because a vector field maps a point to a vector, vector fields are represented by graphing arrows (vectors) at each point.
Checkpoint5.1.1.
Use technology to graph the following vector fields.
Vector fields appear in many applications. The problems below illustrate vector fields with common concepts.
Activity68.
The purpose of this activity is to connect the idea of vector fields to physical examples, to introduce the idea of a vector line integral, and the idea of vorticity.
(a)
Suppose a leaf drops in the stream in Figure 5.1.2 at \((x,y)=(0,\pi)\text{.}\) Sketch the path it will follow. At what point will it end up?
(b)
Suppose a leaf drops in the stream in Figure 5.1.2 at \((x,y)=(0,0)\text{.}\) Sketch the path it will follow. At what point will it end up?
(c)
Suppose a leaf drops in the stream in Figure 5.1.3 at \((x,y)=(0,\pi)\text{.}\) Sketch the path it will follow. At what point will it end up?
(d)
Suppose a neutrally buoyant ball with radius \(0.1\) has its center at \((0,0,0.5)\) in the stream in Figure 5.1.4. What is the force at the top of the ball? What is the force at the bottom of the ball? What will this cause the ball to do?
