91Ó°ÊÓ

In vector calculus, a transformation involves mapping a set of points from one space to another. This is primarily represented by a function that takes inputs as vectors and translates them to new positions. For the exercise in question, the transformation is defined by the function \( T(u, v) = (-u^2 + 4u, v) \). This means that each point \((u, v)\) in the domain \(D^*\) is moved to a new location based on the equations provided in the function.

• The transformation for the \(u\)-component is \(-u^2 + 4u\), which changes depending on the value of \(u\).
• The transformation for the \(v\)-component is simply \(v\), meaning the \(v\)-value remains unchanged.

These transformed points make up what is known as the image of the transformation. By analyzing the transformation, we can understand how the initial rectangle \([0,1] \times [0,1]\) changes shape and orientation in the new space. The task describes how these transformations affect each component, thus helping to find the overall image of \(D^*\). Understanding transformations is key to solving more complex vector calculus problems, as it's the basis for analyzing how shapes and functions behave when manipulated or mapped from one coordinate system to another.

A function is considered one-to-one if it maps every distinct element of its domain to a distinct element in its codomain. For the function to be one-to-one, or injective, different inputs must yield different outputs. In the exercise, we consider whether the transformation \( T(u, v) \) is one-to-one.

• The \(u\)-component's transformation, \(T(u) = -u^2 + 4u\), is not one-to-one. Despite being distinct values, different \(u\) inputs can map to the same transformed output.
• This is due to the quadratic nature of the function forming a parabola, which is not monotonic within the specified range of \([0,1]\).

Because of its parabola shape, values in the function \(-u^2 + 4u\) can repeat themselves, demonstrating why it is not one-to-one over the interval \([0,1]\). Recognizing when a function is one-to-one is fundamental, especially when determining if functions can be inverted or uniquely solved.

Quadratic functions are a cornerstone in mathematics, described as functions of the form \(ax^2 + bx + c\). They characterize parabolic graphs and possess specific traits like an axis of symmetry and a vertex, which can be a maximum or minimum point. In the exercise, the transformation function \(-u^2 + 4u\) alters the \(u\)-component and forms a downward opening parabola.

• This transforms into the format \(-(u^2 - 4u) = -(u-2)^2 + 4\), showing us the maximum point at \(u = 2\).
• Since our domain is restricted to \([0,1]\), the parabola's maximum achievable value is reduced from 4 to 3 as \(u=1\) gives the boundary value 3.

Understanding the behavior of quadratic functions is crucial for interpreting transformations, as they elucidate the non-linear relationships and potential values dictated by their structure. Such insights enable practitioners to anticipate changes during a transformation and analyze how alterations affect the shape and position of images in vector spaces.