5. Let F = (-y, x), and let C be the line segment from (0, 2) to (1,5). (a) Parametrize the line segment into the form r(t) = (x(t), y(t)). Specify the range of t values. (b) Evaluate the line integral So F.dr.

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: Let f(x, y, z) = 5x³ - y³ + z². Find the maximum value M for the directional derivative at the point…

A:

Q: [[(2x + y²) da + 2xy dy], where C is the line segment from (-1,0) to (1, 2) Evaluate in two…

A: The line integral ∫CMdx+Ndy involves the integration over a specific path. In this case, the line…

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: 3. Given F(x,y)=(y,2x), find F-T ds where curve C is the parabola y=x from point (0,0) to point…

A: We use line integration to evaluate this.

Q: F(x, y) = (4x³,3y²) along a path from (0, 0) to (2, 6). Find S F · dř. a) using a parabolic path. b)…

A:

Q: 1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a…

A:

Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: 6. Let f(x, y) = ln(2x - y²). (a) (b) (c) Determine the gradient of f at the point (2, -1).…

A:

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Sketch the curve r(t)=f°i+tj and indicate the orientation. Find r'(t) and sketch r(1) & r°(1).

A: First we will plot the given parameters.

Q: Integrate the function f(r,y, z) = Over the entire 3D coordinate system, i.e., all of R'.

A: We have to integrate the function fx, y, z=1x2+y2+z21+x2+y2+z2 over the entire 3D coordinate system,…

Q: 6. Given f(x, y) = 2x³y=x²y³, a) find the directional derivative of f(x, y) at (1, 1) in the…

A:

Q: 6. Let F(x, y, z)=2xy² + 2yz² + 2x²z. a. Determine the maximum directional derivative of F at the…

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: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: For the function f(x, y) = 2x² – 3xy + 8y² + 2x – 4y + 4, find a tangent vector to the level curve…

A:

Q: Find an equation (x(t), y(t), z(t)) of the line that is the intersection of tangent planes at (1,…

A:

Q: Compute the work done by the force F = (sin(x + y), xy, x²z) in moving an object along the…

A: Given information:Force vector in Newtons.Initial point .Intermediate point .Final point .To find:…

Q: . Evaluate SF.dr: ▪ F = (x+z)i +zj+yk. C is the line from (2,4,4) to (1,5,2).

A: Since all subparts are different and not interlinked so as per our guideline we are supposed to…