Problem 11 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. The vector-valued function $$ \mathbf{F}(x, y)=y \mathbf{i}+x^{2} \mathbf{j}+x y \mathbf{k} $$ is an example of a vector field in the \(x y\) -plane.

Step-by-Step Solution

Answer

False, because the vector function has a non-zero component in the z-dimension ( \mathbf{k}) and hence, is not confined to the xy-plane.

Step 1: Understanding Vector Fields

A vector field in the plane assigns a vector to every point \( (x, y) \) in the plane. Typically, such a vector would have components that are functions of \(x\) and \(y\), expressed as \( \mathbf{G}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \). Here, \(P\) and \(Q\) are both functions of \(x\) and \(y\), and \mathbf{G}\ is two-dimensional.

Step 2: Examine the Given Function

The given function is \( \mathbf{F}(x, y) = y \mathbf{i} + x^2 \mathbf{j} + xy \mathbf{k} \). This function has three components: \( y, x^2, \) and \( xy \). A vector in three-dimensional space is represented as \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), where \(R\) depends on a third dimension (often denoted as \(z\)).

Step 3: Determine the Dimensionality

For a vector field to be in the \(xy\)-plane, it should only have \(i\) and \(j\) components, giving it a two-dimensional presence. The presence of \(\mathbf{k}\) in \(\mathbf{F}(x, y) = y \mathbf{i} + x^2 \mathbf{j} + xy \mathbf{k}\) indicates a third dimension, meaning it is not limited to the \(xy\)-plane.

Step 4: Conclusion: Is it a Vector Field in the Plane?

Since \(\mathbf{F}(x, y)\) includes the \(\mathbf{k}\) component, suggesting a behavior in the z-dimension, it cannot be considered a vector field strictly within the \(xy\)-plane. A true vector field in the plane would lack any \(\mathbf{k}\) component.

Key Concepts

Vector Valued Function

A vector valued function is a mathematical expression that takes one or multiple variables as input and returns a vector output. In the case of the provided function, the input variables are \( x \) and \( y \). The function \( \mathbf{F}(x, y) = y \mathbf{i} + x^2 \mathbf{j} + xy \mathbf{k} \) is an example of such a function.
\( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors that indicate directions in space, often corresponding to the axes of a Cartesian coordinate system.
\( \mathbf{i} \) represents the x-axis, \( \mathbf{j} \) represents the y-axis, and \( \mathbf{k} \) represents the z-axis. The components of the function, \( y \), \( x^2 \), and \( xy \), assign magnitudes to these directions, effectively creating a vector for each point \( (x, y) \).
This function results in a three-dimensional vector because of the presence of all three unit vectors, suggesting influence from variables in three dimensions.

Two-Dimensional Systems

A two-dimensional system in mathematics refers to functions or equations that involve only two variables and operate largely within a plane. In terms of vector-valued functions or vector fields, a two-dimensional representation is characterized by vectors that extend in the \(x-y\) plane with only \(\mathbf{i}\) and \(\mathbf{j}\) components.
When the problem indicates consideration of the \(xy\) plane, it implies the vectors should only translate over this plane without a third-dimensional influence, which is indicated by the absence of a \(\mathbf{k}\) component.

• A true two-dimensional vector field for this scenario would be expressed as \( \mathbf{G}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \).
• Each component \(P\) and \(Q\) relates solely to the variables \(x\) and \(y\), without reference to any other dimensional variable.

The inclusion of \(\mathbf{k}\) in the vector \(\mathbf{F}(x, y)\) implies motion or effects into the third dimension, thereby disqualifying it from being a purely two-dimensional system.

Dimensionality in Mathematics

Dimensionality in mathematics refers to the number of independent directions in a space. It is foundational in understanding where mathematical objects such as points, lines, surfaces, and vectors exist within or interact with a given space.

• One dimension, denoted by variables such as \(x\), describes linear spread.
• Two dimensions, with variables like \(x, y\), describe planar spread, confined to the \(xy\)-plane.

For three-dimensional space, represented by variables \(x, y, z\), objects have depth in addition to length and width. In vector-valued functions, dimensionality is determined by the axes along which the vectors have components.
The presence of a \(\mathbf{k}\) component in the vector \(\mathbf{F}(x, y)\) signifies interaction within this three-dimensional space, which includes the z-axis. This suggests a multi-dimensional view beyond a simple plane, thus the vector field described here reaches beyond a two-dimensional system.