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Show that the curve with parametric equations $ x = t \cos t $, $ y = t \sin t $, $ z = t $ lies on the cone $ z^2 = x^2 + y^2 $, and use this fact to help sketch the curve.

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02:13

Wen Zheng

Calculus 3

Chapter 13

Vector Functions

Section 1

Vector Functions and Space Curves

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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"].

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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.

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Show that the curve with p…

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Show that the curve $\math…

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Let $\mathcal{C}$ be the c…

So we're gonna make use of the fact that we have X equals because sign t co sign of T that is y equals t sign of t NZ just equals t. So knowing that we can adjust our parametric equations and we know that Z squared is going to be equal to X squared plus y square, that is our cone equation, Then we can plug in the different values we have, which gives us t squared equals t co sign t squared plus t sign t squared. That's just plugging in our values of X, y and Z and putting in the t values. So then we know that this is just going to equal t squared equal to, um we can factor out a t squared once we do all this so we get t squared equals C squared times co sign square T plus sine square t We know that coastline square T plus sine squared T is always going to equal one So we have a t squared equals C squared. Um, and since both sides are equal, since this is a tautology or something, that's absolutely true. We know that the equation is true. so the curve will lie on the cone. We know that if Z equals K, then since this is a cone like this of Z is constant, then our traces are going to be circles like that. We know if y equals K, so why is constant? Then we would end up getting hyperbole, as so we let y equal constantly get hyperbole. And also, since it's symmetrical, we know that if we let X equal k, we will also get hi purple eyes. And since the Parametric equations trace how to spiral with decreasing radius, um, as he gets closer to zero, um, and then it's an increasing radius as T is greater than zero. So we see when t increases, we get the increasing radius like that.

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