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A level curve of \(f(x,y)\) is a curve on the domain that satisfies \(f(x,y) = k\). It can be viewed as the intersection of the surface \(z = f(x,y)\) and the horizontal plane \(z = k\) projected onto the domain.
Draw a circle in the xy-plane centered at the origin and regard it is as a level curve of the surface
- Section 2.2: Parametrized surfaces
- 1. Planes.
- B(t) = T(t) × N(t) the bi-normal vector
There is a different, fundamentally different way to describe a surface. It is called parametriza-tion of a surface. This is achieved with a vector-valued function hx(u,v),y(u,v),z(u,v)i r (u,v) = . It is given by three functions x(u,v),y(u,v),z(u,v) of two variables. Because two parameters u and v are involved, the mapr is often called uv-map. If ...
Parametric:r (s,t) = OP + sv + tw Implicit: ax + by + cz = d. We can change from parametric to implicit using the cross productn =v ×w . We can change from implicit to parametric by finding three points P,Q,R on the surface and forming
If we differentiate T(t) T(t) = 1, we get T′(t) T(t) = 0 and see that N(t) is perpendicular to T(t). The three vectors (T(t),N(t),B(t)) are unit vectors orthogonal to each other. Here is an application of curvature: If a curver (t) represents a wave front andn (t) is a unit vector normal to the curve at r (t), thens (t) =r (t)+n (t)/κ(t) defines a ...
Level curves. Before the advent of calculus, a curve is usually de ned through level sets: (in the plane) as level sets: f(x; y) = c; (in the space) as intersection of surfaces (intersection of level sets): f(x; y; z) = c1; g(x; y; z) = c2: Example 1. A circle in R2 is represented as. (x ¡ a)2 + (y ¡ b)2 = r2:
level curve. For example, f(x;y) = 4x2 + 3y2 = 1 is an ellipse. Level curves allow to visualize functions of two variables f(x;y). LEVEL SURFACES. We will later see also 3D ana-logues: if f(x;y;z) is a function of three variables and cis a constant then f(x;y;z) = cis a surface in space. It is called a contour surface or a level surface.
A level curve, or surface, is a set on which f is constant. If you are on a level curve, and you want to stay on that curve, which way should you travel? Using the mountain analogy, determine the direction of maximum slope and turn 90°.
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Level Curves and Contour Plots. Level curves and contour plots are another way of visualizing functions of two variables. If you have seen a topographic map then you have seen a contour plot. Example: To illustrate this we first draw the graph of z = x2 + y2.
