What is the difference between the following definitions of Vector Functions and Parametric Curves?
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The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space as the input varies throughout it's domain values of t.
Vector Functions:
Vector functions are given by a: I $rightarrowmathbb{R}^n$ with the domain I, an interval of $mathbb{R}$, and the range being a subset of two or three dimensional space. For example, with $n = 3$, with respect to the fixed orthonormal basis ${bf{e}}_1$, ${bf{e}}_2$ and ${bf{e}}_3$. The vector ${bf{a}}(t)$ can be written as $${bf{a}}(t)=a_1(t){bf{e}}_1 + a_2(t){bf{e}}_2 + a_3(t){bf{e}}_3.$$
Parametric Curves:
A parametric curve is a function x : I →$mathbb{R}^n$, x : t $mapsto$ x(t) where I is an interval. We will consider $n = 2$ (two-dimesional parametric curves, or plane curves) and $n = 3$ (three-dimensional parametric curves, space curves). Some definitions further require the function x to be differentiable. As the parameter t varies in the interval I, the point with position vector x(t) traces out a curve.
functions vector-spaces vectors parametric plane-curves
asked Jan 12 at 14:01
ImranImran
31417
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add a comment |
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The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space as the input varies throughout it's domain values of t.
Vector Functions:
Vector functions are given by a: I $rightarrowmathbb{R}^n$ with the domain I, an interval of $mathbb{R}$, and the range being a subset of two or three dimensional space. For example, with $n = 3$, with respect to the fixed orthonormal basis ${bf{e}}_1$, ${bf{e}}_2$ and ${bf{e}}_3$. The vector ${bf{a}}(t)$ can be written as $${bf{a}}(t)=a_1(t){bf{e}}_1 + a_2(t){bf{e}}_2 + a_3(t){bf{e}}_3.$$
Parametric Curves:
A parametric curve is a function x : I →$mathbb{R}^n$, x : t $mapsto$ x(t) where I is an interval. We will consider $n = 2$ (two-dimesional parametric curves, or plane curves) and $n = 3$ (three-dimensional parametric curves, space curves). Some definitions further require the function x to be differentiable. As the parameter t varies in the interval I, the point with position vector x(t) traces out a curve.
functions vector-spaces vectors parametric plane-curves
asked Jan 12 at 14:01
ImranImran
31417
$endgroup$
$begingroup$
What's "an interval of $mathbb R^n$"?
$endgroup$
– Calvin Khor
Jan 12 at 14:09
$begingroup$
That was an error. I've corrected it to $mathbb{R}$. Thanks.
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– Imran
Jan 12 at 14:17
1
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In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $mathbb R^n to mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same
$endgroup$
– Calvin Khor
Jan 12 at 14:26
add a comment |
$begingroup$
The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space as the input varies throughout it's domain values of t.
Vector Functions:
Vector functions are given by a: I $rightarrowmathbb{R}^n$ with the domain I, an interval of $mathbb{R}$, and the range being a subset of two or three dimensional space. For example, with $n = 3$, with respect to the fixed orthonormal basis ${bf{e}}_1$, ${bf{e}}_2$ and ${bf{e}}_3$. The vector ${bf{a}}(t)$ can be written as $${bf{a}}(t)=a_1(t){bf{e}}_1 + a_2(t){bf{e}}_2 + a_3(t){bf{e}}_3.$$
Parametric Curves:
A parametric curve is a function x : I →$mathbb{R}^n$, x : t $mapsto$ x(t) where I is an interval. We will consider $n = 2$ (two-dimesional parametric curves, or plane curves) and $n = 3$ (three-dimensional parametric curves, space curves). Some definitions further require the function x to be differentiable. As the parameter t varies in the interval I, the point with position vector x(t) traces out a curve.
functions vector-spaces vectors parametric plane-curves
asked Jan 12 at 14:01
ImranImran
31417
$endgroup$
The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space as the input varies throughout it's domain values of t.
Vector Functions:
Vector functions are given by a: I $rightarrowmathbb{R}^n$ with the domain I, an interval of $mathbb{R}$, and the range being a subset of two or three dimensional space. For example, with $n = 3$, with respect to the fixed orthonormal basis ${bf{e}}_1$, ${bf{e}}_2$ and ${bf{e}}_3$. The vector ${bf{a}}(t)$ can be written as $${bf{a}}(t)=a_1(t){bf{e}}_1 + a_2(t){bf{e}}_2 + a_3(t){bf{e}}_3.$$
Parametric Curves:
A parametric curve is a function x : I →$mathbb{R}^n$, x : t $mapsto$ x(t) where I is an interval. We will consider $n = 2$ (two-dimesional parametric curves, or plane curves) and $n = 3$ (three-dimensional parametric curves, space curves). Some definitions further require the function x to be differentiable. As the parameter t varies in the interval I, the point with position vector x(t) traces out a curve.
functions vector-spaces vectors parametric plane-curves
functions vector-spaces vectors parametric plane-curves
asked Jan 12 at 14:01
ImranImran
31417
asked Jan 12 at 14:01
ImranImran
31417
edited Jan 12 at 14:24
Imran
asked Jan 12 at 14:01
ImranImran
31417
asked Jan 12 at 14:01
ImranImran
31417
asked Jan 12 at 14:01
ImranImran
31417
31417
$begingroup$
What's "an interval of $mathbb R^n$"?
$endgroup$
– Calvin Khor
Jan 12 at 14:09
$begingroup$
That was an error. I've corrected it to $mathbb{R}$. Thanks.
$endgroup$
– Imran
Jan 12 at 14:17
1
$begingroup$
In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $mathbb R^n to mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same
$endgroup$
– Calvin Khor
Jan 12 at 14:26
add a comment |
$begingroup$
What's "an interval of $mathbb R^n$"?
$endgroup$
– Calvin Khor
Jan 12 at 14:09
$begingroup$
That was an error. I've corrected it to $mathbb{R}$. Thanks.
$endgroup$
– Imran
Jan 12 at 14:17
1
$begingroup$
In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $mathbb R^n to mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same
$endgroup$
– Calvin Khor
Jan 12 at 14:26
$begingroup$
What's "an interval of $mathbb R^n$"?
$endgroup$
– Calvin Khor
Jan 12 at 14:09
$begingroup$
What's "an interval of $mathbb R^n$"?
$endgroup$
– Calvin Khor
Jan 12 at 14:09
$begingroup$
That was an error. I've corrected it to $mathbb{R}$. Thanks.
$endgroup$
– Imran
Jan 12 at 14:17
$begingroup$
That was an error. I've corrected it to $mathbb{R}$. Thanks.
$endgroup$
– Imran
Jan 12 at 14:17
1
1
$begingroup$
In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $mathbb R^n to mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same
$endgroup$
– Calvin Khor
Jan 12 at 14:26
$begingroup$
In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $mathbb R^n to mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same
$endgroup$
– Calvin Khor
Jan 12 at 14:26
add a comment |
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$begingroup$
What's "an interval of $mathbb R^n$"?
$endgroup$
– Calvin Khor
Jan 12 at 14:09
$begingroup$
That was an error. I've corrected it to $mathbb{R}$. Thanks.
$endgroup$
– Imran
Jan 12 at 14:17
1
$begingroup$
In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $mathbb R^n to mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same
$endgroup$
– Calvin Khor
Jan 12 at 14:26