To determine Find the value of unit tangent vector ( T ) , the principal unit normal vector ( N ) and the curvature ( κ ) . Explanation Given information: The expression of plane curve is, r ( t ) = t i + a cosh ( t a ) j , a > 0 (1) Formula used: Write a general expression to calculate constant vector v . v = d r d t (2) Here, d r d t is the value of change in plane curve. Write a general expression to calculate the unit tangent vector. T = v | v | (3) Write a general expression to calculate the principal unit normal vector. N = ( d T d t ) | d T d t | (4) Here, d T d t is the value of change in unit tangent vector. Write a general expression to calculate the curvature. κ = 1 | v | | d T d t | (5) Calculation: Differentiate equation (1) with respect to t . d r ( t ) d t = ( 1 ) i + a ( 1 a ) sinh ( t a ) j = i + sinh ( t a ) j Substitute i + sinh ( t a ) j for d r ( t ) d t in equation (2) to find v . v = i + sinh ( t a ) j Take the magnitude of above equation to find | v | . | v | = | i + sinh ( t a ) j | = 1 2 + ( sinh ( t a ) ) 2 = 1 + sinh 2 ( t a ) = cosh 2 ( t a ) { ∵ cosh 2 θ = 1 + sinh 2 θ } | v | = cosh ( t a ) Substitute i + sinh ( t a ) j for v , and cosh ( t a ) for | v | in equation (3) to find T . T = i + sinh ( t a ) j cosh ( t a ) = 1 cosh ( t a ) i + sinh ( t a ) cosh ( t a ) j = sech ( t a ) i + tanh ( t a ) j { ∵ sech θ = 1 cosh θ tanh θ = sinh θ cosh θ } Differentiate above equation with respect to t . d T d t = − 1 a sech ( t a ) tanh ( t a ) i + 1 a sech 2 ( t a ) j Take magnitude for above equation to find | d T d t |

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Chapter 13.4, Problem 15E

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Find the value of unit tangent vector (T), the principal unit normal vector (N) and the curvature (κ).

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Chapter 13.4, Problem 14E

Chapter 13.4, Problem 16E

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Chapter 13 Solutions

EP THOMAS'CALCULUS,EARLY TRANS.-MYLAB

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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