) Let F(x, y) = (3x² + y²)i + 2xyj. (a) Test if F is a conservative vector field. (b) If F is conservative, find a potential function for it. (c) Evaluate the line integral Sc(3x² + y²)dx + 2xydy along the arc given by x = 2 cos t, y = 3 sint, 0 < t < 5. Use both the single integrals and the Green's

Q: Find the directional derivative of the scalar field ¢ = xªy + 3xz at the point (1, −1, 4) along the…

A:

Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: Given r(t) be a vector function that satisfies: r'(1)= T'(1)= 1) Determine the unit tangent…

A: Given, r'(1)=&lt;-3,0,4&gt; T'(1)=&lt;2,2,1&gt;

Q: X p(x, y, z)=xz²+sin (y) ex

A:

Q: Find the curl of the vector field F = (7x sin(y), 3y cos(x)). curl F

A: The given vector field F=7xsiny, 3ycosx. The objective is to find the curl F. Use the derivative…

Q: Determine whether the vector field F(x, y) = 4 sin(y)i + (-8y + 4x cos(y)) is conservative or not.…

A:

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder + (y − 1)² = 1 (a)…

A: We will use the basic knowledge of vector calculus and differential calculus to answer all three…

Q: Find the equation of the gradient vector field for the function f(x, y) = – læ² + 1y and then sketch…

A: The gradient vector field of given function is -2x i + 2y j

Q: = (4x° – 7y)i + (9y' + 7x – 7)j %3D - ndary of the parallelogram in the following figure (note the…

A: Introduction: Let D be associated open, a merely connected region with a piecewise swish,…

Q: Suppose F(x, y) = x² + y² and C is the line segment from point A = (1, -1) to B = (3,-5). (a) Find a…

A: Given          Let       (a)   The equation of the line segment from A(1, -1) to B(3, -5) is,,…

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

A:

Q: A) Consider the vector-field F(x, y) = 2xy³i+ 3y²x²j. 1. Show that F is conservative. 2. Find a…

A: The given vector field is  F⇀(x,y)=2xy3i⇀+3y2x2j→ We have to show that F→ is conservative

Q: 6) Let f(xxx)=3xy²-x² be a potential function of a fluid velocity field. a) Find the magnitude of…

A:

Q: Let C be the curve which is the union of two line segments, the first going from (0, 0) to (1, 4)…

A: As per Bartleby guidelines I have authority to answer only one question. If you want other…

Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

A:

Q: The gradient of f (x, y) = x sin (y²) is the vector field F(x, y) = at (2,√7/6) and ending at…

A: The given scalar function is, f(x,y)=xsiny2. Gradient of this function is F=siny22xycosy2. The…

Q: 3. (a) Given the vector valued function r(t) = 4cos nti + 4 sin ntj + tk. Find the position vector,…

A:

Q: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

A:

Q: Let the velocity vector be v(t) = (6 cos(3t), 8 exp(2t), 3t2). Compute the acceleration vector a(t).

A:

Q: A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The…

A:

Q: Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential…

A:

Q: The gradient of f(x, y) = x sin (y²) is the vector field F (x, y) = at (2,√π/6) and ending at…

A: Here we are given that the gradient of f(x,y)=xsin(y2) is the vector field F(x,y)=sin(y2)2xycos(y2)

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

A:

Q: Consider the vector field F(2. v) = (--2) i + (8ry° tan ' y) j. i + (8ry² – tan ' y) j. a) Is F…

A: Consider the vector field

Q: Consider the function g defined by 1 g(x, y) = cos (Ta y) + %3D log3(x – y) Do as indicated.…

A: please comment if you need any clarification. If you find my answer useful please put thumbs…

Q: for the attched r(t): 1) how do I find r'(t) 2) sketch the position vector r(0) 3) sketch the…

A: given that r(t)=e2ti+etj. we need to  : (a) find r'(t) r'(t) is the first derivative of the function…

Q: 3. Find the line integral fF.dr where C' is the circle of radius 5 in the xz-plane oriented…

A: The given vector field is .Find the value of line integral  as follows.The parametric surface can be…

Q: Find the curl of the vector field F (6x sin(y), 2y cos(x)). curl F =

A:

Q: Find the curl of the vector field F = (5y cos(x), 4x sin(y)) curl F -

A: