Answered: • • Let > be a potential for the vector… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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Let > be a potential for the vector field F = (−2 y³, −6 xy² − 4 z³, −12 yz² + 4 2). Then the value of
sin((-1.63, 2.06, 0.57) – (0,0,0)) is
-
0.336
-0.931
-0.587
0.440
0.902
0.607
-0.609
0.146

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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 VI
Let (4xy-3x²z²)₁ + (2x²)Ĵ − 2x³ zk = (i) Find a scalar potential for F. (x, y, z) = - note: Answer should have "+C" in it, where C is some constant (ii) Find the work done in moving from (1,1,0) to (2, −1, 1) by the above force field. W= (iii) Solve the differential equation (4xy - 3x²²)dx + 2x²dy - 2x³zdz = 0 ☐ = K, where K is a constant
The answer for number 1 is D Could you show me why
Find the directional derivative of the scalar field = x³y + 4xz at the point (1,-1, 2) along the direction vector (2, -1, −2):
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x - y, z + y +7,z²) and the net is decribed by the equation y = √1-x²-2², y 20, and oriented in the positive y-direction. (Use symbolic notation and fractions where needed.) 1.45-1 yas
Find the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):
Please don't provide hand written solution
Let 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.
2. Let x' = [5, 1, 3] and y' = [−1, 3, 1]. (a) Find the inner product, y'x. (b) Find x - 2y. (c) Find the length of x, Lx. (d) Find the length of y, Ly. (e) Find the unit vector of x (i.e., the unit vector with the same direction as x). (f) Find the unit vector of y.
(b) Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.
(a) y7 + x] (b)-y7 + x] (c) x7 +y] (d)-y7 + (x+7)7 10 (e)-y7 + (x-7) 7 10 (f) (x−y) 7+ (x−y) 7 IV V III In WE V II

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