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I am just reviewing some assumptions in Parametric representations

The book says we assume 3-d curve has non-vanishing tangent vector. Why do we need to assume this

Simply if we take $R^3$ then assume $x=f(t), y=g(t), z=h(t)$

clearly the tangent vector $dx/dt, dy/dt,dz/dt$ are all not zero at the same time as its not possible so why do we need to assume this assumptions as its an intrinsic property of $R^3$ i don't think we need to assume this.

to put it plainly.

logically we can't draw a curve which is parallel to x, y and z axis at same time. this is an intrinsic property of every curve in 3d so why do we need to explicitly assume this.

Mr. Math
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1 Answers1

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I just learned that there are curves which have a zero tangent vector.

basically curves which are stationary at one or more points.

if there is no other parameterization where the curve is non stationary then the curve has zero tangent vector at the point.

hence the assumption was necessary when dealing with PDE

Mr. Math
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