How many elements does the vector space V over the field F={0, 1} have, if its dimension is dimV=8? (a) 8. (b) 16. (c) 64. (d) 256. (e) Infinitely many
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How many elements does the
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- Ellipse : a+=1 2 -V2)4. In Parts (a)-(b), you are given a vector field F. Use graphical reasoning to decide whether the div F(1,1) is positive, negative, or zero. Justify your answer with a complete sentence explanation. (a) (b)Q5: a) Show that the vector field F = (x² - y² + x)i- (2xy + y)j, is irrational.
- Consider vector field the デ- 4Sn (x?)? + (3x-+ + (3x* + 3e)} line ntegral SF.d? Find the F. d? where with vertices C is the rectangle 6,0), 4,0), (o,3), and, (413) oriented counter clockwise.Which of the following expressions are meaningful (where F is a vector field and f is a function)? Of those that are meaningful, which are automatically zero? (a) div(∇f ) (b) curl(∇f ) (c) ∇curl(f ) (d) div(curl(F)) (e) curl(div(F)) (f) ∇(div(F))2. Let r = √x² + y² + z². (a) For n ≥ 1 and r > 0, express V(r-n) in terms of r and the radial vector field er. (b) For r> 0, express Vlnr in terms of r and the radial vector field er. (c) Taking A as in question 1, show that for r> 0 we have A(-¹) = 0.
- ·SoF F.Tds for the vector field F = x²i+yj along the curve x = y2 from (1,1) to (4,-2). Evaluate SoF F.Tds= (Type an integer or a simplified fraction.)Q5) If the vector field T = (axy + Bz³)a, + (3x² – yz)a, + (3xz² – y)a, is irrotational, determine a, ß, and y. Find V-T at (2, -1, 0). (Note: irrotational means the curl equal to zero)[SADT8] If Aand Bare vector fields, prove the following: V (A · B) = (B -V) A + (A · V) B+B × (V×A)+A × (V×B).
- Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector field. (5x + 7y, 8x + 5y, 0) = 1 + where the first vector in the sum is curl free and the second is divergence free. (For this problem, enter your vectors with angle-bracket notation: , not in ijk- notation.)The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a) Is div(F) positive, negative, or zero at P? Explain. div(F) is ---Select--- because the vectors that start near P are ---Select--- those that end near P. (b) Determine whether curl(F) = 0. If not, in which direction does curl(F) point at P? curl(F) + 0. At P the curl(F) points in the direction of positive x. curl(F) + 0. At P the curl(F) points in the direction of negative x. curl(F) + 0. At P the curl(F) points in the direction of positive y. curl(F) + 0. At P the curl(F) points in the direction of negative y. curl(F) + 0. At P the curl(F) points in the direction of positive z. curl(F) + 0. At P the curl(F) points in the direction of negative z. curl(F) : = 0Consider the scalar field f(x, y, z) = 2x - y² + 3z2 Determine V Vfl1.-1.1) II. VxVfl(1.-1.1)
