Written Homework for June 24, 2024, Vector Calculus II First Name Last Name OSU ID Grading Some written homework problems are graded for correctness. To earn full credit for each problem you need to clearly and neatly show your work or provide a written explanation when appropriate. Partial credit will be awarded for logical, organized progress toward a solution when work is required and shown. Correct answers with no supporting work will not receive full credit. Problems 1. What topics from Vector Calculus I did you personally find the most interesting? What topics did you personally find the most challenging? 2. Consider the parallelogram formed by two vectors ⃗u and ⃗v . (a) Show that the height of the parallelogram is ∥⃗v ∥ sin θ, where θ is the angle between ⃗u and ⃗v . (b) Show that the area of the parallelogram is ∥⃗u × ⃗v ∥. 1 3. Consider the slanted box (parallelepiped) formed by three vectors ⃗u, ⃗v , and w. ⃗ (a) Show that the height of the box is ∥w∥ ⃗ cos θ, where θ is the angle between ⃗u × ⃗v and w. ⃗ (b) Show that the volume of the box is |w ⃗ · (⃗u × ⃗v )|. (This quantity is called the scalar triple product of ⃗u, ⃗v , and w.) ⃗ (c) Suppose that ⃗u = ⟨u1 , u2 , u3 ⟩, ⃗v = ⟨v1 , v2 , v3 ⟩, and w ⃗ = ⟨w1 , w2 , w3 ⟩. Show that the volume of the parallelepiped is given by   w w2 w3  1   det  u1 u2 u3  . v1 v2 v3 2 4. Consider the space curve ⃗r(t) = ⟨x(t), y(t), z(t)⟩ = ⟨3t, 4t, 5 ln sec t⟩. r (a) Find d⃗ = r⃗′ (t) = ⟨x′ (t), y ′ (t), z ′ (t)⟩. dt (b) Find the arc length of ⃗r(t) from t = 0 to t = π3 . 5. Consider the scalar function f (x, y) and a level curve f (x, y) = c. Another way to think about this level curve is to view y as a function of x, so we could write f (x, y(x)) = c. This helps when taking the (not partial) derivative of f with respect to x. (a) Differentiate both sides of c = f (x, y) with respect to x. Be sure to use the multivariate chain rule for the right side. Conclude that dy fx =− , dx fy where fx and fy denote the partial derivatives of f . 3 dy (b) Use implicit differentiation (part (a)) to find dx for the circle x2 y 2 = 1. (c) Draw the circle x2 y 2 = 1 and a few tangent lines. Do the slopes agree with the answer from part (b)? 6. Consider the scalar function φ(x, y) and the level curve φ(x, y) = c. From the previous problem dy x . we know that dx = −φ φy (a) Conclude that ⟨φy , −φx ⟩ is tangent to the level curve at (x, y). (b) Compute the dot product of ∇φ and ⟨φy , −φx ⟩. 4