QuadricCylindText - Cylinders, Quadric Surfaces, and...

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Cylinders, Quadric Surfaces, and Slicing John E. Gilbert, Heather Van Ligten, and Benni Goetz Now, fnally, we are almost ready to develop calculus oF Functions oF several variables. Let’s start by trying to understand the graph oF some particular Functions z = f ( x, y ) and, more generally, relations f ( x, y, z ) = 0 . Some examples oF relations f ( x, y, z ) = 0 that will later become crucial make a good starting point. Planes: we’ve seen that a plane can be described by a Linear Equation Ax + By + Cz = D where the vector n = a A, B, C A oF coe±cents is normal, i.e. , perpendicular, to the plane. Linear here means that all the variables have power one . The graph oF x + y + z = 2 is shown in pink to the right below. To see why, let’s start with the coordinate planes and axes as shown to the leFt below. Recall that a plane is determined by three points on it or by any pair oF lines in the plane. ²or 3 points, take the intercepts x = 2 , y = 2 , and z = 2 on the coordinate axes. Do you see them on the right? What’s a natural pair oF lines in the pink plane to use? x + y + z = 2 How about taking the intersection oF x + y + z = 2 with the y = 0 plane? This is the line x + z = 2 ; see it to the right? ²or the other line we could take the intersection y + z = 2 with the x = 0 plane. This idea oF slicing, oFten by the coordinate planes, will be a crucial tool For us.
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It all starts with a double cone, the graph of the quadratic equation z 2 = x 2 + y 2 to the left below. A
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QuadricCylindText - Cylinders, Quadric Surfaces, and...

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