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Course Notes for Multivariable Calculus

Remkes Kooistra

Section 9.1 Vector Fields

Subsection 9.1.1 Definition

So far in this course and Calculus III, the multivariable functions I’ve introduced were scalar fields \(f: \RR^n \rightarrow \RR\) and parametric curves \(\gamma: [a,b] \rightarrow \RR^n\text{.}\) Parametric curves have a single variable of input and scalar fields have a single variable of output. Now I am finally going to give a complete treatment of function which have multiple variables for input and for output.
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That said, there are two examples of the functions with multiple inputs and output that have already been consider: linear transformation in linear algebra and change of coordiantes functions. Both these examples treated function \(\RR^m \rightarrow \RR^m\) as transformations of space. The analysis of linear transformation in linear algebra implicit used this interpretation throughout: what does the linear algebra do to \(\RR^n\text{?.}\) The determinant calculate the change on size and orientation. Invertibility wondered if the changes could be undone. The classification of types was all about rotations, projects and the like things that happen to the ambient space. Change of variables functions were also about changing space by changing the coordinate descriptions. The Jacobians even used the same determinant idea to talk about the change in (local, infinitesimal) size under a change in variables.
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Interpreting functions \(F: \RR^n \rightarrow \RR^m\) as transformations of space is very useful and valuable. It connects well with linear algebra, where matrices are thought of as transformations. It serves multiple integration by allowing coordinate transformations. However, it is not the only conceptual way to understand such functions. I am now going to introduce a new interpretation of these functions, and along with the new interpretation, a new name for these functions.
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Definition 9.1.1.

A function on a region of \(S \subset \RR^n\) which outputs vectors is a vector field. If the output are vectors with \(m\) components, I can write the function as \(F: S \rightarrow \RR^m\text{.}\) By notational convention, I’ll use captial letters for vector fields and lower case letter for scalar fields.
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If \(F: S \rightarrow \RR^m\) is a vector field, then I can write \(F = (F_1, F_2, \ldots, F_m)\) in terms of its components. Each component \(F_i\) is itself a scalar field, has partial derivatives and gradients, and is subject to all the tools already defined for scalar fields.
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The use of the word ‘field’ matched the scalar fields of Calculus III. A scalar field assigned a scalar to every point in a region \(S \subset \RR^m\text{.}\) In familiar \(\RR^3\text{,}\) scalar fields could assign temperature, pressure, density, concentration and similar scalars to point. Now, a vector field assigns a vector to each point in a reagion \(S \subset \RR^n\text{.}\) This is the new interpretation. The ambient space is unchange, but each point in the ambient space is assigned a vector which has some meaning.
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What does it mean to assign a vector to each point of space? The vector could represent a force (say a gravitational or electromagnetic force) that applies at any point in space. If could represented the local movement or acceleration of a fluid (wind speed/direction and ocean current speed/direction are a vector fields). It could represent the rate and direction of heat flow or diffusion of concentration. It could be the gradient of any scalar field (temperature gradient, pressure gradient, concentration gradient, etc.) The key thing, in all of these, is that the measurement assigns a vector fo each point: something with a direction and a magnitude.
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The vector of a vector field is always a local direction vector. It is a vector as if the point in question is the origin, pointing out in some direction with some magniture from the current point in space.
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Subsection 9.1.2 Examples

To visualize vector fields, I can draw arrows on a region in \(\RR^2\text{.}\) I’ll do this with two examples.
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Example 9.1.2.

Figure 9.1.3. The Vector Field \(F(x,y) = (0,1)\)
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Consider the constant vector field \(F(x,y) = (0,1)\) on \(\RR^3\text{,}\) as seen in Figure 9.1.3. The vector \((0,1)\) is a unit vector in the vertical y-axis direction in \(\RR^2\text{.}\) This vector fields associates that unit vertical vector to each point in \(\RR^2\) (as a local direction). At each point, the vector points vertically upward from that point. If this field represented a fluid flow, the fluid would be flowing upwards (in \(y\)) at a uniform pace everywhere, following these local direction vectors.
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Example 9.1.4.

Figure 9.1.5. The Vector Field Describing Gravitational Force
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An excellent example of a vectors field is the force of gravity, shown in Figure 9.1.5. The magnitude of the force of gravity per unit mass due to a mass \(M\) at the origin is a scalar field. \((x,y,z)\) is:
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\begin{equation*} f = \frac{MG}{x^2+y^2+z^2} \end{equation*}
However, the force itself also includes direction, so it is a vector field.
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\begin{equation*} F = \frac{MG}{(x^2+y^2+z^2)^{\frac{3}{2}}} (-x,-y,-z) \end{equation*}
The direction \((-x,-y,-z)\) is back towards the origin. I can recover the magnitude by taking the length of the vector.
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\begin{equation*} |F| = \frac{MG}{(x^2 + y^2 + z^2)^\frac{3}{2}} \sqrt{x^2 +y^2+z^2} = \frac{MG}{x^2 +y^2+z^2} = \frac{MG}{r^2} \end{equation*}
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Through the discussion of vector fields, which last for most of the remainder of the course, I will rely heavily on two applications: fields of force and fields describing fluid flow. These two applications are fundamental to the theory. Between them, they can help ground the difficult abstration of the discipline. For fields of force, gravity is often the first example indeed, I used it above. However, electromagnetic force will end up being the more important example. Historically, the development of the theory of vector fields was driven by 19th century attempts to understand electricity and magnestism. Many of the basics of gravitational action can be understood with scalar calculus, as Newton did in his physics. However, eletricity and magnetism are difficult to even start describing without vector fields.
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Subsection 9.1.3 Basic Operations on Vector Fields

Vector fields produce vectors, one for each point in the domain. Since the output of the field is a vector, all the operations of vectors apply. Let me review these operations.
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Assume that \(F\) and \(G\) are two vector fields on some region \(S \subset \RR^n\text{.}\) Assume that \(f\) is a scalar field on the same region. All these operations are the ordinary vector operations done individually at every point in the domain \(S\text{.}\)
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  • The length of a vector field, which is the length of each output vector, is a scalar field.
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    \begin{equation*} |F| \end{equation*}
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  • If a vector field never has length zero, then I can define the field of unit direction of the vector field. This new field gives the direction of movement at each point in the vector field, but erases the maagnitude.
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    \begin{equation*} \frac{F}{|F|} \end{equation*}
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  • The sum or different of two vectors fields is another vector field.
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    \begin{equation*} F \pm G \end{equation*}
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  • The dot product of two vector fields is a scalar field.
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    \begin{equation*} F \cdot G \end{equation*}
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  • The scalar product of a scalar field with a vector field is another vector field.
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    \begin{equation*} fF \end{equation*}
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  • If \(S \subset \RR^3\text{,}\) then the cross product of two vectors fields is another vector field.
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    \begin{equation*} F \times G \end{equation*}
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