By means of Stokes' Theorem, find | F. dŘ 3.x, on the ellipse x² + y? = 1, z = F = xỉ+ (x + y)3+ (x + y + z). The ellipse is traced counterclockwise if we look back at it from the direction of the positive z-axis.

The curve in the plane defined by the equation (*) (x^2 + xy + y^2 = 7) is a rotated ellipse, shown in the figure. a) Use implicit derivation to find dy/dx expressed by x and y. (b) Show that the curve passes through the point (2,1) and find the equation of the tangent line to the curve at the point (2,1).

Find a parametric function for the intersection of r – 3y + 4z = 0 and a? + y² = 9.

The plane x+y+2z=4 intersects the paraboloid z=x^2+y^2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin

Let C be the curve defined by the following system of equations: √x+6y²-2²=0 3y = z