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- F(x, y) = x²yi + 3y³x²j Find the integral curves for the vector field а. —6у + 1 = C 6xy2 ob. -6ху — 1 C 6xy2 c. -6ху + 1 = C 6x²y2 O d. -ху + 6 C x?y2Find the directional derivative of the scalar field = x³y + 4xz at the point (1,-1, 2) along the direction vector (2, -1, −2):Q4 A) Calculate the divergence of the vector field F=y³i + xyj B) Find the Curl of F(xy.z)=3x²i + 2zj - xk.
- 2. For the vector field V = (7x³ + 2y2²)i + (4x²z - 7y²+3)j + (4y³ + x² − 32²) k find the divergence V.v and the curl V x v.Find the curl of the vector fieldF (6y cos(x), 7x sin(y)) curl F =Find the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):
- ·SoF F.Tds for the vector field F = x²i+yj along the curve x = y2 from (1,1) to (4,-2). Evaluate SoF F.Tds= (Type an integer or a simplified fraction.)Q2. (a) Given a vector field, F = 2xyi + (x² + 3y²z)j+ (y³ + 2z)k. Find the potential function of F and evaluate it at the point A(1,3, 2).A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x-y, z + y + 3, z²) and the net is decribed by the equation y = √1-x²-2², y ≥ 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)