Show the vector field V = (8xy³ z, 12x²y² z, 4x²y³) is irrotational, and find its potential. Then, by using the Path Independence Integral, and the Fundamental Theorem of Calculus for Vector Fields evaluate the line integral along the helix parameterized by F(t)= (2 cos t, 2 sin t, t) from (2, 0, 0) to (1, √3, π/3)

Find a potential function for the vector field .F(x,y)=<16xy+8y-2,8x^2+8x>

Draw the direction field for dy dx = yln(x + y) using MATHEMATICA!

In the images below, the vector field F is shown by the blue arrows and the curve Cis shown in red. Indicate the value of F ·dT. 1. /, F ·dÌ = 0 2. ,F ·dT > 0 3. F dT < 0 4. The value of F ·dT cannot be determined with only this information. X