Problem 2 (10pt). Let F = xyzi+yzj + zk. Compute the divergence and curl of F.
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- Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).5. Calculate the divergence of F = (7x², 9y, -52).
- (2) Consider the function f: Nx N→Q given by f(x, y) = = your claim. X y + 1 Find the image of f and proveEvaluate ∫ ▽f*dr Where f(x,y,z)=cos(πx)+sin(πy)-xyz and C is any path that starts at (1,1/2,2) and ends at (2,1,-1)What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy and Vx = -Uy) do not hold?
- 35. Evaluate F.dr: (a) F = (x+z)i +zj+yk. C is the line from (2,4,4) to (1,5,2). (b) F= x²i+zsin(yz)j + y sin(yz)k. C is the curve from A4(0,0,1) to B(3,1,2) as shown below. B A (c) F = yi-xj+zk. C is the circle of radius 3 centered on the z-axis in the plane z=4 oriented clockwise when viewed from above. (d) F = 4x³i + (x+y)j. C is the curve y=sin(2x) from (0,0) to (7/2,0). (e) F = (-y³ + sin(x²))ī + (x³ − ln(y² +1)) j . C is the circle of radius 5 centered at (0,0) in the .xy- plane oriented counterclockwise.#2. Let flz) = (x, y) +iv(x,y) be an analytic function in a domain D. If a zeck. y) + b v(x, y) = C in D are real Constants not all 0. where a, b and c Prove that flz) is constant in D.If F(t) = 2xyi + xyz2j + ex+yk, then what is the divergence of F? Select one: A. 2yi + xz2j + ex+yk B. 2y + xz2 C. 2yi + xz2j + zex+yk D. 2y + xz2 + ex+y
