Problem 11 - Advanced Engineering Mathematics

Question

Find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ z=x^{2}+y^{2}, y=x ; x=t $$

Step-by-Step Solution

Answer

The vector function is \( \mathbf{r}(t) = \langle t, t, 2t^2 \rangle \).

Step 1: Understand the Given Surfaces

We are given two surfaces: one is the paraboloid given by \( z = x^2 + y^2 \) and the other is the plane \( y = x \). We want to find the intersection of these two surfaces.

Step 2: Use the Parameterization

The problem indicates to use the parameter \( x = t \). This means we will express other variables in terms of \( t \). The parameterization gives us \( x = t \), and since \( y = x \), we have \( y = t \).

Step 3: Substitute into the Paraboloid Equation

Substitute \( x = t \) and \( y = t \) into the equation for the paraboloid: \( z = x^2 + y^2 = t^2 + t^2 \). Simplifying this, we get \( z = 2t^2 \).

Step 4: Form the Vector Function

Now that we have all components in terms of \( t \): \( x = t \), \( y = t \), and \( z = 2t^2 \), we can write the vector function for the curve \( C \) as \( \mathbf{r}(t) = \langle t, t, 2t^2 \rangle \).

Step 5: Sketch the Curve

To sketch the curve \( C \), note that it traces along the plane where \( y=x \), and the \( z \)-value is determined by \( z=2t^2 \). This is a curve on the paraboloid, following a diagonal (\( y=x \)) path from the vertex outward, with \( z \) increasing quadratically.

Key Concepts

Vector Function

In vector calculus, a vector function plays a crucial role in describing curves within spaces of any dimension. A vector function essentially assigns a vector to each value of a parameter, usually time, marked as parameter \( t \). By doing this, it maps out a path or trajectory in space.

For example, the vector function \( \mathbf{r}(t) = \langle t, t, 2t^2 \rangle \) describes a curve in three-dimensional space. The components \( \langle x(t), y(t), z(t) \rangle \) indicate the position of a point at any time \( t \).

• The component \( x(t) = t \) tells us how the position in the \( x \)-direction changes along the curve.
• The component \( y(t) = t \) mirrors this movement in the \( y \)-direction.
• The component \( z(t) = 2t^2 \) describes how the position in the \( z \)-axis changes.

This means as \( t \) varies, the vector traces out the path of the curve in the three-dimensional space.

Parameterization

Parameterization is a technique to express curves or surfaces in terms of parameters. This is very useful in vector calculus because it allows us to describe complex geometric shapes with simpler mathematical equations.

In our context, parameterization involves assigning a specific parameter value to the \( x \)-coordinate, using \( x = t \). Through parameterization, we translate a multivariate system to one that depends on a single variable. This transformation is done as follows:

• Set \( x = t \) from the parameterization given.
• Since \( y = x \), translate this to \( y = t \).
• From the paraboloid equation \( z = x^2 + y^2 \), substitute \( t \) for both \( x \) and \( y \) to find \( z = 2t^2 \).

This transformation simplifies the process of finding the vector function by reducing the complexity of solving for multiple variables.

Intersection of Surfaces

When two surfaces intersect, they create a curve. Finding the intersection of surfaces is often necessary to understand how different geometric bodies interact in space.

To find this curve of intersection, we must satisfy the equations of both surfaces. In this exercise, we are dealing with a paraboloid defined by \( z = x^2 + y^2 \) and a plane defined by \( y = x \). Combining these:

• The plane condition \( y = x \) simplifies the process since we only need one parameter \( x = t \) to describe both \( x \) and \( y \).
• The paraboloid equation becomes \( z = 2t^2 \) through substitution, simplifying it to a function of \( t \).

Rendering both surface equations in this unified parameter space allows us to construct a single curve, expressed as \( \mathbf{r}(t) \). This curve represents the intersection line on which both the given surfaces meet.

Paraboloid

A paraboloid is a particular type of surface generated by revolving a parabola around its axis of symmetry. This results in a three-dimensional shape that opens upwards (or downwards), much like a bowl.

The paraboloid in the exercise is defined by the equation \( z = x^2 + y^2 \). This equation describes a surface where any point \( (x, y) \) on the plane corresponds to a height \( z \) calculated as the sum of the squares of \( x \) and \( y \).

• The term \( x^2 + y^2 \) forms circles when \( z \) is constant. Thus, the horizontal cross-sections of a paraboloid are circular.
• As \( z \) increases, the radius of these circles increases, forming the familiar bowl shape.

In three-dimensional modeling, paraboloids are common, appearing in telescopes, antennas, and arches. Understanding this structure helps in visualizing how curves behave on surfaces like the one formed by the intersection with the plane described in the exercise.