2. Consider the triangle {(x, y)| x, y > 0, 2x + y < 1}. Let C be the boundary of this triangle oriented in the anticlockwise direction. Find y dx + (x² + y) dy (i) using the definition of line integral directly and without using Green's theorem. (ii) using Green's theorem.

Q: Evaluate the integral |(2e' – 2x) dy +(2xy +1) dx using Green's theorem where C is C 64 ans:-- 3…

A: Using Greens theorem the integral over a closed path can be converted into a corresponding surface…

Q: Evaluate ʃC F dr where C is the parabola y = x2 from (0, 0) to (2, 4) and F (x, y) =&lt; −ysinx,…

A: Given a curve C is a parabola from (0, 0) to (2, 4)

Q: 5. Find an equation plane to the surface (u, v) = (3 cos(u) sin(v), 2 sin(u) sin(v), cos(u)) at the…

A: First, we find the point on the surface where the tangent plane would be drawn. Next, we calculate…

Q: Find a point on the surface x+4y--z² =1 where the tangent plane is parallel to the plane x+4y- z =…

A: we need to find gradient of given surface to know about direction of normal vector to tangent plane

Q: 14. In an experiment of exercise and health, the pretest and posttest scores for the control group…

A: The effect of the exercise is the change in the value due to exercise. The change can be calculated…

Q: r(u, v)=(w+v)i(u-v)j+v²k at the point P(2, 0, 1)

A:

Q: At which point does the parametric curve given by x = 2 cost y = 2 sin t z = 2t - π/3 cross the…

A:

Q: (x² + xy) dx + (y² - xy) dy; C: the line y = x from the origin to the point (2, 2).

A:

Q: = . Find an equation of the tangent plane to the parametric surface R(u, v) 0≤u≤, at the point…

A:

Q: 3. Let f(x, y) = xy+ y². Find the equation of the tangent line in parametric form in the x-direction…

A: Solve the following

Q: Find an equation of the tangent plane to the parametric surface R(u, v) = ((u - sin u) cos v, (1 -…

A:

Q: B) r(t) = (4 cos t, 4 sin t, 16)

A: We consider x=rcos(theta) and  y=rsin(theta)  Therefore r^2=x^2+y^2

Q: SHOW COMPLETE SOLUTION, MAKE THE SOLUTION CLEAR. THANK YOU!

A: Step 1: Step 2: Step 3: Step 4:

Q: Find the tangent plane to the equation z = 6y cos(4x – 2y) at the point (2,4,24) Z =

A: It is given that z=6ycos4x-2y and the point is 2,4,24. Recall the fact that if the equation of the…

Q: Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in…

A:

Q: Evaluate This Integral if curve C consists of curve C₁ which is a parabola y=x² from point (0,0) to…

A:

Q: Consider the paraboloid given by the equation: z = x^2 + y^2 Find the equation of the tangent…

A: at point (1, 2, 5)at point (1, 2, 5)

Q: ôw ôw + = 0 for w = Ax, y), x = u – v, and y = v – u and demonstrate the ди ду 6. Show that result…

A: