6. Find the circulation and outward flux of the velocity field = sin yi + cos y) over the square cut from the first quadrant by the lines and y = .
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- 5. The electric potential is V volts at any point (x, y) in the xy plane and V = e-2" cos(2y). Distance is measured in feet. (a) Find the rate of change of the potential at the point (0, r) in the direction of the unit vector cosTit sinr î. COSExpress the flowing equation in Cartesian equation r = 1-cos eFind the directional derivative of ø = x² + y² + z² at point (1, 2, 1) for a direction determined by dx = 2dy = -2dz.
- Find the direction field for the equation: y' = x + sin(y) + cos (x) at the point A = (0, pi3)4. The particle moves along a plane curve y = e, where x and y are measured in meters. It has a constant speed v= 12 m/s. Determine the x and y components of acceleration at y = 1 m. Also determine the tangential and normal components of acceleration at this instant.+4y= 16t (Laplace)
- Calculate directional derivatives and gradients in three dimensions. Find an equation for the tangent plane to the surface z + 4 = xy° cos(2) at the point (4, 1, 0).Sketch the curve with the vector equation r(t) = cos(t)i − cos(t)j + sin(t)k. Show the direction of increasing t with an arrow drawn on your curve.find Pn (Q'U R).
- bo Find (xT x)" xT y where, b2 40 57 112 45 54 118 50 54 128 55 60 121 60 66 126 65 59 136 70 61 144 75 58 142 80 59 149 85 56 165 S SINARLINEshow full and complete procedure HANDWRITTEN onlyA particle moves in the xy− plane according to the law x=at, y=bt2, where a>0, b>0. Determine the particle’s trajectory y(x) and sketch its graph. Determine the speed of the particle as a function of time. Find the angle ϕ between the velocity vector and the x−axis.