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Try to sketch by hand the curve of intersection of the parabolic cylinder $ y = x^2 $ and the top half of the ellipsoid $ x^2 + 4y^2 + 4z^2 = 16 $. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

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$x=t, y=t^{2}, z=\frac{1}{2} \sqrt{16-t^{2}-4 t^{4}}, \quad-1.37073 \leq t \leq 1.37073$

03:27

Wen Zheng

Calculus 3

Chapter 13

Vector Functions

Section 1

Vector Functions and Space Curves

Campbell University

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University of Michigan - Ann Arbor

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Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

03:19

Try to sketch by hand the …

03:12

02:40

Find a vector function tha…

12:38

The ellipsoid $4 x^{2}+2 y…

02:52

for this problem we have that y equals X squared and X squared plus four y squared plus four z squared equals 16. So this is the top half of the Ellipse oId, um is going to be the positive square root. So we have is, um if we subtract over this right here and this right here we get that four z squared equals 16 minus x squared, minus four y squared. Um, so then Z is going to equal one half the square root of 16 minus x squared, minus four y squared. Then looking at the two equations, we can see that why is the same thing can be represented by X? So what this is going to suggest is we can come up with a Parametric representation. So if we let x equal t since y equals X squared, that means that why will equal t squared and then Z which equals this whole thing right here. We can now, right as being equal to one half 16 minus T squared, minus four teas to the fourth. So based on that, we see that the value in the square root is greater than or equal to zero. So that means that we're limited two negative 13707 less than or equal to T, which is less than or equal to 1.3707 So we have our X component of the Parametric equation, or white component of the Parametric equation rz component of the Parametric equation, and then we have what T is limited to.

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