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PILOT Calculus III Fall 2015 Problem Set 6 1. Find the arc length of the given curve on the specified interval. a. ) (2 cos 𝑡, 2 sin 𝑡, 𝑡) for 0 ≤ 𝑡 ≤ 2𝜋 b. ) ((𝑡 2 + 1) cos 𝑡, (𝑡 2 + 1) sin 𝑡, 2√2𝑡) for 0 ≤ 𝑡 ≤ 1 3 c. ) ( 2 cos 3𝑡, 2 sin 3𝑡, 2𝑡 2 ) for 0 ≤ 𝑡 ≤ 1 2. Let 𝐜: [𝑎, 𝑏] ⟶ ℝ3 be an infinitely differentiable path (derivatives of all orders exist). Assume 𝐜 ′ (𝑡) ≠ 𝟎 for any t. the vector 𝐜′(𝑡)/‖ 𝐜 ′ (𝑡)‖ = 𝐓(𝑡) is tangent to c at 𝐜(𝑡), and, becaute ‖𝐓(𝑡)‖ = 1, T is called the unit tangent to c. a. Show that 𝐓 ′ (𝑡) ∙ 𝐓(𝑡) = 0. (Hint: Differentiate 𝐓(𝑡) ∙ 𝐓(𝑡) = 1.) b. Write down a formula for 𝐓 ′ (𝑡) in terms of c. 3. Let 𝐜(𝑡) be a given path, 𝑎 ≤ 𝑡 ≤ 𝑏. Let 𝑠 = 𝛼(𝑡) be a new variable, where α is a strictly increasing C1 function given on [a,b]. For each s in [α(a), α(b)] there is a unique t with α(t) = s. Define the function 𝐝: [𝛼(𝑎), 𝛼(𝑏)] → ℝ3 by d(s) = c(t). a. Argue that the image curves of c and d are the same. b. Show that c and d have the same arc length. 𝑡 c. Let 𝑠 = 𝛼(𝑡) = ∫𝑎 ‖𝐜 ′ (𝜏)‖𝑑𝜏. Define d as above by d(s) = c(t). Show that 𝑑𝐝(𝑠) ‖ ‖=1 𝑑𝑠 The path 𝑠 ⟼ 𝐝(𝑠) is said to be an arc-length reparameterization of c. 4. Sketch the given vector field or a small multiple of it. a. ) 𝐅(𝑥, 𝑦) = (𝑦, −2𝑥) b. ) 𝐅(𝑥, 𝑦) = (2𝑦, 𝑥) 5. Find a function 𝑓: ℝ3 → ℝ such that 𝐅 = ∇𝑓 for a. ) 𝐅(𝑥, 𝑦, 𝑧) = (𝑦𝑧, 𝑥𝑧, 𝑥𝑦) b. ) 𝐅(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 𝑧) 6. Let 𝐅(𝑥, 𝑦, 𝑧) = (𝑥𝑒 𝑦 , 𝑦 2 𝑧 2 , 𝑥𝑦𝑧) and suppose 𝐜(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) is a flow line for F. Find the system of differential equations that the functions 𝑥(𝑡), 𝑦(𝑡), and 𝑧(𝑡) must satisfy. 7. Let 𝐅(𝑥, 𝑦, 𝑧) = (𝑒 𝑥𝑧 , sin(𝑥𝑦) , 𝑥 5 𝑦 3 𝑧 2 ). a. Find the divergence of F. b. Find the curl of F. 8. Let 𝐫(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 𝑧) and 𝑟 = √𝑥 2 + 𝑦 2 + 𝑧 2 = ‖𝐫‖. Prove the following identities: 1 𝐫 a. ∇ ( ) = − 3 , 𝑟 ≠ 0; and, in general, ∇(𝑟 𝑛 ) = 𝑛𝑟 𝑛−2 𝐫 and ∇(log 𝑟) = 𝐫/𝑟 2 𝑟 1 𝑟 b. ∇2 ( ) = 0, 𝑟 ≠ 0; and, in general, ∇2 (𝑟 𝑛 ) = 𝑛(𝑛 + 1)𝑟 𝑛−2 𝑟 9. Is it possible to solve the system of equations 𝑥𝑦 2 + 𝑥𝑧𝑢 + 𝑦𝑣 2 = 3 𝑢3 𝑦𝑧 + 2𝑥𝑣 − 𝑢2 𝑣 2 = 2 for 𝑢(𝑥, 𝑦, 𝑧), 𝑣(𝑥, 𝑦, 𝑧) near (𝑥, 𝑦, 𝑧) = (1,1,1), (𝑢, 𝑣) = (1,1)? Compute 𝜕𝑣 ⁄𝜕𝑦 at (𝑥, 𝑦, 𝑧) = (1,1,1).