1. In each part, you're given a vector field ( langle P(x, y), Q(x, y) angle ). Determine the signs of the partial derivatives ( P_{x}, P_{y}, Q_{x} ), and ( Q_{y} ). (The sign may be different at different points.) Hint: In (a) and (b), three of ( P_{x}, P_{y}, Q_{x} ), and ( Q_{y} ) are 0 everywhere. (a) (b) (c) 2. 2-dimensional curl. Suppose each vector field in #1 represents the velocity of water in a shallow pool. (a) If you put a tiny paddlewheel in the vector field #1(b) at (1, 1), which way would it spin? What if you put the paddlewheel at (-1, 1)? (b) What if you instead put the tiny paddlewheel in the vector field #1(a) at (1, 1)? (c) Suppose you put the tiny paddlewheel in the vector field #1(b) at (0.5, 0.5). How does its spinning compare to if you put it in the same vector field at (1.5, -1)? (d) Suppose you have a new vector field ( vec{F}(x, y)=langle P(x, y), Q(x, y) angle ). What quantity could describe the spin of a tiny paddlewheel at a point in this vector field?
Added by Jovany S.
Close
Step 1
- \( P(x, y) \) is constant and positive, \( Q(x, y) = 0 \). - Partial derivatives: \( P_x = 0 \), \( P_y = 0 \), \( Q_x = 0 \), \( Q_y = 0 \). Show more…
Show all steps
Your feedback will help us improve your experience
Andreas Papavassiliou and 64 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
1. Recall if F is the curl of another vector field, then ∇ · F = 0 everywhere. In this problem, you will investigate whether or not F(x,y,z) = x/(x^2 + y^2 + z^2)^{3/2} i + y/(x^2 + y^2 + z^2)^{3/2} j + z/(x^2 + y^2 + z^2)^{3/2} k is the curl of another vector field. (a) Compute the divergence of F. (b) Let S1 and S2 be the upper and lower halves of the unit sphere, both endowed with upward pointing normal vectors. Argue that if F = curl G for some continuously differentiable vector field G, then flux of F across S1 and the flux of F across S2 are equal. (c) With S1 and S2 as above, compute the integrals ∬_{S1} (F · n) dσ and ∬_{S2} (F · n) dσ directly. Is F the curl of another vector field or not?
Adi S.
Melissa M.
For each of the following, answer true or false. (a) For any smooth vector field F in R^3, we have div(curl(F)) = 0. (b) The directional derivative of a function f(x, y), at a point (x0, y0), in the direction of some vector v, can always be computed by the simple formula Dvf = ∇f(x0, y0) · v. (c) The curvature of any closed curve is always positive at every point on the curve. (d) Any vector field F is the gradient of some function f. (e) For any twice differentiable function f(x, y) with continuous second-order partials, the mixed partials are equal, i.e. ∂^2f/∂y∂x = ∂^2f/∂x∂y.
Madhur L.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD