7. Evaluate the line integral v dr where C is the curve formed by the intersection of the cy 2²+²=4 and the plane z+2x-y-3, travelled counterclockwise as viewed from the positive z-axis, and v is the vector function v=zi+(z-y)j +yzk.

Q: 3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at…

A: Step 1: Step 2: Step 3: Step 4:

Q: Find a vector function that represents the curve of intersection of the paraboloid z = 5x² + 2y² and…

A: The paraboloid z=5x2+2y2 and the cylinder y=3x2. We have to find a vector function that represents…

Q: Let L be the line passing through the point P(−4, −3, 4) with direction vector d=[-3, 3, 0]T, and…

A:

Q: Find a vector function that represents the cur ve of intersection of the paraboloid z a + mnd the…

A: Given that, the paraboloid is z=x2+y2 and the cylinder is x2+y2=22. The projection of the of the…

Q: 3. Find a vector function that represents the curve of intersection of the paraboloid z =. and the…

A: To find the vector function that represents the curve of intersection of the paraboloid z=x2+y2, and…

Q: Find a vector equation with parameter t for the line of intersection of the planes x + y + z = 4 and…

A: To find the equation of line with parameter t of the intersection of the planes  x + y + z =4 and…

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Consider the following position function in the plane: ř (t) = (t3 – 3t)i + 3t?j. %3D For this…

A: Position vector of any parametric curve r(t) can be given by a vector joining origin with that point…

Q: 2. Find a vector that is perpendicular to the surface x² + y² + z² = xyz + 8 at the point (1,2,3).

A: Given that x²+y²+z²=xyz+8 At (1,2,3)

Q: Let S be the surface obtained by revolving the plane curve 2x=sqrt(9−y2) about the y-axis.Obtain a…

A: Given- Let S be the surface obtained by revolving the plane curve 2x = 9 - y2 about the y-axis. To…

Q: Consider the following function. T: R? - R, T(x, y) = (2x?, 3xy, y²) Find the following images for…

A:

Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: Show that the curve with vector equations ~r(t) = is a plane curve

A:

Q: 8. A line initiated from (1,0,2) and parallel to a vector i+j+k what is the parametric equation of…

A: we have to find equation of line in 8.

Q: B-For the surface z² = x² + y² find the directional derivative at the point (√2, √2,-2) in the…

A:

Q: Let L be the line passing through the point P=(4, -3, 1) with direction vector -[-4, 1, 1], and let…

A:

Q: 6. (a) Show that the curve represented by the vector function F(t) = (5t, 3t – 4, 2t² – 6) is at the…

A: