Concept explainers
(a) Find a slope field whose
(b) Prove that if
(c) Find an equation that implicitly defines the integral curve through
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- Match each of the functions (a)-(c) with the corresponding gradient vector field from those in (I)-(VI). Enter your answer as upper case roman numerals: I, II, III, etc. (a) f(x, y) = x cos(y) (b) f(x, y) = (x - 2)²(y-2)/5 (c) f(x, y) = sin(x) · cos(y) (click on an image to enlarge it) IV II V II VIarrow_forwardF = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?arrow_forwardShow that (vector)F = <ln(y) + y/x, ln(x) + x/y > is a gradient field. Then find a potential function f for (vector) F.arrow_forward
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- 7. Find the outward flux of the field F(x, y) = -yi + x² j across the closed counter-clockwise curve consisting of the top semi-circle of + y? = 4 followed by the straight line from (-2,0) to (2,0).arrow_forwardFind the flux of the field F =(x + y)i -(x2+y2)j outward across the triangle with vertices (1, 0), (0, 1), (-1, 0).arrow_forwardLet q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.arrow_forward
- 3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part of this problem.) (a) Plot a contour map of f. (b) Find the gradient Vf. (c) Plot the gradient vector field Vf. (d) Explain how the contour map and the gradient vector field are related. (e) Plot the flow lines of Vf. (f) Explain how the flow lines and the vector field are related. (g) Explain how the flow lines of Vf and the contour map are related.arrow_forwardThe gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² )i O II. (4xy)i -(2x2 -3y? )i O II (4xy)i +(4x2 -3y² )i OV. (4xy)i +(2x2 -3y² )iarrow_forward(b) Show that the vector field, F3 yz î + zx ŷ + xy 2 can be written both as the gradient of a scalar and curl of a vector. Find the scalar and vector potential for this function.arrow_forward
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