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- Integrate ƒ(x, y, z) = x - 3y2 + z over the line segment C joining the origin to the point (1, 1, 1)Calculate the curve integral ∫c ( 3x2 - 3yz + 2xz )dx + ( 3y2 - 3xz +z2 )dy + ( 3z2 - 3xy + x2 +2yz )dz , where ? is any curve path from (-1,2,3) to (3.2,-1).A point moves along the curve of intersection of the paraboloid z=x^2+5y^2 and the plane x=3. At what rate is z changing with y when the point is at (3,-1,14)?
- determine a and b as to make the cylinder y^2=4ax orthogonal to the ellipsoid at point (1,2,1)Suppose ƒ(1, 2) = 4, ƒx(1, 2) = 5, and ƒy(1, 2) = -3. Find an equation of the plane tangent to the surface z = ƒ(x, y) at the point P0(1, 2, 4).Show that the line normal to the surface xy + z = 2 at the point (1, 1, 1) passes through the origin.
- 5) Consider four points A, B, C, and D whose geometrical locations correspond to the corners of a square with sides of length 1 mm. Calculate the potential differences (in mV) VAB, VBA, VAC, VCA, VAD, VDA, VBC, VCB, VBD, VDB, VCD, VDC between the points in a uniform electric field of 3 V/m parallel to the two sides (and perpendicular to the other two) of the square. (Show your work)Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.Double integrate under z=xy, above the triangle with vertices (0,1),(0,4),(1,1).