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Chapter 9: Problem 40

What is meant by the trace of a surface? How do you find a trace?

Short Answer

Expert verified
The trace of a surface is the curve that results from the intersection of the surface with a plane. To find a trace, one must determine the equation of the plane with which the surface will intersect, and resolve these equations simultaneously.

Step by step solution

01

Understanding the concept of Trace

In mathematics, the trace of a surface is the curve that results from the intersection of the surface with a plane. It can be visualized in 3D space as the path you would track if you were to 'slice' the surface with a blade (the plane). Often the plane is chosen to be parallel with one of the Cartesian planes in order to make the plot easy to investigate.
02

Finding the Trace

To find a trace, determine the equation of the plane with which the surface will intersect. This plane could be inclined at any angle but is typically chosen such that the plane is parallel to one of the three major planes (xy-plane, yz-plane, xz-plane). Once the plane equation is decided, resolve the equation of the surface and the plane simultaneously. This intersection gives us the trace of a surface on the plane.

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Most popular questions from this chapter

If the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) has the same magnitude as the projection of \(\mathbf{v}\) onto \(\mathbf{u}\), can you conclude that \(\|\mathbf{u}\|=\|\mathbf{v}\|\) ? Explain.

Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\mathbf{i}, \quad \mathbf{v}=\mathbf{i} $$

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