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The view of the trefoil knot shown in Figure 8 is accurate, but it doesn't reveal the whole story. Use the parametric equations$$ x = (2 + \cos 1.5t) \cos t $$$$ y = (2 + \cos 1.5t) \sin t $$$$ z = \sin 1.5t $$to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $ xy $-plane has polar coordinates $ r = 2 + \cos 1.5t $ and $ \theta = t $, so $ r $ varies between 1 and 3. Then show that $ z $ has maximum and minimum values when the projection is halfway between $ r = 1 $ and $ r = 3 $. When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)
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Vector Functions and Space Curves
Johns Hopkins University
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University of Nottingham
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"].
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.
The view of the trefoil kn…
You will use a CAS to expl…
The problem is use parametric equations x, equal to 2 pasco sine 1 point in y is equal to 2 pas cosine 1.50 times sine t z is equal to sine 1.5 t to sketch the curve by hand as leaved from off, with gaps indicating wise curve passive. Over itself, starting by showing that the production of the curve of x y plane has polar coordinates, are equal to 2 plus cosine 1.5 t, and that is equal to t and then show that z has its maximum and minimum values. When the projection is halfway. Between 1 and 3 point first look at this graph in figure 8 here so figure 8. This is now well sketch this graph by hanso. It looks like this and then the projection of the curve on to x y planeshalfe, if they use polar, coordinates, r, is equal to x, squared plus y square root of x, squared plus y squared. So this is equal to 2 plus cosine 1.5. To here we use the fact sine t square plus sine t square is equal to 1 and theta here is equal to we have tangent, theta is equal to y over x. This is equal to tangent, went theta is equal to t so or lyrics between 1 and the 3 then show that g has maximum and minimum values, while the projection is halfway between r equal to 1 and r equal to 3. Look at it. The equation of d, o t is equal to sine 1.5 to 1. Cosine 1.5 t is equal to 0. We know sine 1.5 t his maximum and minimum values 1 or negative 1 point for in this case, r is equal to 2 point. So this is half way between r equal to 1 and r equal to 3 poi, and now we use the computer to jathagraph with different veal points. Lookahere this is a gap, so we change is real points. Weight,
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