Question 2 (30 points) Integrate f(x, y,2) xzv2-z2 - y2 over the path C, which consists of two curves, C1 and C2 from (...
(1) Integrate f(x, y,z)+Vy - z2 over the straight line segment path from (0,0,0) to (1,1,1) (2) Consider the field F (2xyz+2,x2z, x2y), (a) (b) (c) Show that the field is conservative. Find a potential function for the field. Find the work the field does on an object that follows the path consisting of the line segment from (0,0,0) to (1,2,2), followed by the line segment from (1,2,2) to (2,4,3) Find the work done by the field ß-(x, 3y,-5z) along...
and C2 in the xy-planedefined by the parametric equations Consider trajectories on two curves C1:x=t?, y=t? - <t<«. C2: x = 3t, y=t?, - <t<mo. These two trajectories are known to *intersect* at exactly two points. The origin (0,0) is one of them. And there is another one, which we'll call P. Find Pand select the choice below which gives the slope of the tangent line to the first curve at the point P. Note that only ONE of the...
1. (2 points) Find F dF if curl(F) 3 in the region defined by the 4 curves and C4 Ci F . d7 where F(x,y,z)-Wi +pz? + Vi> and C consists of the arc of the 2. (2 points) Evaluate curve y = sin(x) from (0,0) to (π, 0) and the line segment from (π,0) to (0,0). 4 3 3. (2 points) Evaluate F di where F.y,(ry, 2:,3) and C is the curve of intersection of 5 and y29. going...
Question 2. Consider a surface S in the 0 plane with three smooth boundary curves C1, C2, and C3 as shown in the diagram. Each curve is parametrised so that it is traversed in the direction shown by the arrows For a smooth vector field A(x, y, z) you are given the following results: Ca Adr =-3 C2 0.5 2.0 1.5 -0.5 (a) What is the value of the surface integral ▽ × A. ds. if we assume by convention...
→ (1 point) Let Vf-6xe-r sin(5y) +1 5e* cos(Sy) j. Find the change inf between (0,0) and (1, n/2) in two ways. (a) First, find the change by computing the line integral c Vf di, where C is a curve connecting (0,0) and (1, π/2) The simplest curve is the line segment joining these points. Parameterize it: with 0 t 1, K) = dt Note that this isn't a very pleasant integral to evaluate by hand (though we could easily...
(1 point) Let Vf =-8xe-r sin(5y) 20e-x. cos(Sy) j. Find the change inf between (0,0) and (1, π/2) in two ways vf . dr, where C is a curve connecting (0,0) and (1.d2). (a) First, find the change by computing the line integral The simplest curve is the line segment joining these points. Parameterize it: with 03t s 1, r(t)- so that Icvf . di- Note that this isn't a very pleasant integral to evaluate by hand (though we could...
13. (10 points) (a): Find the line integral of f(x, y, z) = x+y+z over the straight-line segment from (1,2,3) to 0,-1,1). (b): Find the work done by F over the curve in the direction of increasing t, where F=< x2 + y, y2 +1, ze>>, r(t) =< cost, sint,t/27 >, 0<t<27.
Please solve these three questions! (1) Length of graphs a) Let a path C be given by the graph of y g(x), a 3 < b, with a piecewise C1 function g : [a, b - IR. Show that the path integral of a continuous function f: IR2- R over the path C is b) Let g : [a, b] - IR be a piecewise C1 function. The length of the graph of g on (t, g(t)). Show that [a,b]...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
answer all parts please except A if you cannot (6) Consider the vector field F(x,y)-《22, 3y). A path is closed if it ends whiere it starts Consider the 3 closed paths starting and ending at (3,0): C1 the circle of radius 3 centered at the origin, C2 the ellipse with equation 2 +3y2-9, and Cs the flat linear path going to -3 and then going straight back. (a) Use GeoGebra to plot the vector field F (b) For each, parametrize...