Theorem 7.5.4. The curvature of a smooth space curve defined by the vector valued function R(t) is given by k(t): X ||R' (t) × R"(t) | ||R' (t) ||³

Q: a. Show that n(t) = -g'(t)i + f(t)j and - n(t)=g'(t)i-f'(t)j are both normal to the curve r(t) =…

A: Using n(t) = - g'(t)i + f(t)j and - n(t) = g'(t)i – f'(t)j are both normal to the curve r(t) =…

Q: 3. Find the curvature of the curve given by 7(t) = (t, t², t) nt (-1, 1, –1).

A: Given curve is We know that curvature to given curve at point (-1,1,-1) , that is at t=-1 is

Q: I need help with this problem and an explanation for the solution described below. (Calculus 3:…

A: Step 1:Step 2:Step 3:

Q: 2. (18 points) Compute the curvature (t) and the torsion r(t) of the following vector function: r(t)…

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Q: (ii)If a = a(s)is a regular differentiable curve , then show that :- (a) kt = -T · B', (b)t = B · N…

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Q: Let C be the space curve with the parametrization r(t) = (cost, sint, In (cost)) for - < t < 4. (a)…

A: Bartleby Guideline : “Since you have posted a question with multiple sub parts, we willprovide the…

Q: Let x(t) be a Bézier curve and the tangent vector x' (t) is computed. What does knowing that x'(0) =…

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Q: 4. Consider the vector function v(t) = (t, t², t³). (a) (10 points) Find the unit tangent vector…

A: ###(a)To find the unit tangent vector T(t), the unit normal vector N(t), and the binormal vector…

Q: Find T, N, and к for the plane curve r(t) = 9t i + 9 In (cos t) j, - T(t)=(☐)i+(☐)j N(t) = ( ☐ ) i +…

A: Step 1:

Q: ||r'(t) × r"(t)|| ||r'(t)||* 3. Use the formula k(t) to find the curvature of the space curve at t =…

A: To find The curvature of the space curve

Q: s question is differential geometry. Prove that the coordinate lines of a su

A: (1).To prove that the coordinate lines of a surface patch are lines of curvature if and only if…

Q: Let r(t) = w +tv be the parametrization of a line. (a) Show that the are length function s (t) = |…

A: The parametric equation of the line in vector form has a standard expression r=x+ty. Here, x is the…

Q: 4. Assume that the curvature of a path described by the vector-valued function r(t) can be ||r' (t)…

A: Question 4:  STEP 1: Define the vector function:For a curve given by y=f(x), we can express it as a…

Q: 3. Consider the circle as a vector-valued function r(t) = cos(t)i + sin(t)j. Calculate limt→+∞ r(t)…

A: Find the limit

Q: 1. Let F(x, y) = (1 — y, ½x³ +e³²), and let C with the position vector r be the portion of the…

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Q: Q. find the curvature K of each plane curve traced out by the vector function r=r(t) = (3t – t3)i +…

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Q: for the attched r(t): 1) how do I find r'(t) 2) sketch the position vector r(0) 3) sketch the…

A: given that r(t)=e2ti+etj. we need to  : (a) find r'(t) r'(t) is the first derivative of the function…

Q: Evaluate [ f (x,y,z) ds given that f(x,y,z) =, and the curve C is ř(t) = 2ti +t²j+÷tk (1<t<∞) %3D…

A: Given: f(x,y,z)=x2xy+6yz r→(t)=2t i→+t2 j→+13t3 k→   1≤t&lt;∞

Q: Let C be a plane curve given by the parametric equations x = f(t), y = g(t), a ≤t≤b. Suppose that C…

A: Let a curve C be parametrized by x= f(t) and y = g(t),

Q: If L is the tangent line to the curve 7 (t)= (2t-1, t, t-2) at (3, 4, 2) and the point where L…

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Q: Let f(x, y) = g(x) + h(y) where g(x) and h(y) are positive. If g(xo) ≥ g(x)and h(yo) ≥ h(y) then…

A: Thus the given FIRST statement is TRUE.