91影视

Use Exercise 52 to show that $\frac{d\mathbf{N}}{ds}=-k\mathbf{T}+味\mathbf{B},$ where 魏 is the curvature. (Hint: Differentiate N = B 脳 T with respect to arc length.)

Every description of the DNA molecule says that the strands of the helices run in opposite directions. This is meant as a statement about chemistry, not about the geometric shape of the double helix. Consider two helices

$$\begin{gathered} h_1\left(t\right)=鉄�\cos 鈦�t,\sin 鈦�t,伪t鉄�and \\ {h}_2\left(t\right)=鉄�\sin 鈦�t,\cos 鈦�t,伪t鉄� \end{gathered}$$

(a) Sketch these two helices in the same coordinate system, and show that they run geometrically in different directions.

(b) Explain why it is impossible for these two helices to fail to intersect, and hence why they could not form a configuration for DNA.

Find all points of intersection between the graphs of the vector functions , and find the acute angle of intersection of the curves at those points.

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Let $\mathbf{r}\left(t\right)=鉄�x\left(t\right),y\left(t\right)|$be a vector-valued function defined on an open interval containing the point $t_0$. Prove that r(t) is continuous at $t_0$if and only if $x\left(t\right)$and $y\left(t\right)$are both continuous at $t_0$.

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54鈥�56.

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54鈥�56.

Find all points of intersection between the graphs of the vector

functions in Exercises, and find the acute angle of

intersection of the curves at those points.

and

Prove that the graph of the vector function $\mathbf{r}\left(t\right)=鉄�t\sin 鈦�t,t\cos 鈦�t,t鉄�,\text{where}t鈮�0$, is a conical helix by showing that it lies on the graph of the cone described by $z=\sqrt{x^2+y^2}$

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Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54鈥�56.