Show that the vector field sin 2y x² Ĥ(x, y) = (y) = (₂ Y x² + 1 is conservative by finding all its potential functions. " -1 tan ¹ x + 2 cos 2y X - In y
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- Prove that the vector given by (sin y sinh x + cos y cosh x) i + (cos y cosh x – sin y sinh x) j is a gradient. Also, find the function having the given vector as its gradient.Find the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):Define two vector functions (t) 9 sin(t)+7 cos(t)] + (t³ - 15) k = = (t) 7 sin(t)+9 cos(t)] + tk = . Compute (t) (t) =
- Express the vector field D = ( x2 + y2 )-1 ( xax + yay ) in cylindrical components and cylindrical variables.Find a potential function for the vector field .F(x,y)=<16xy+8y-2,8x^2+8x>Let F = (41 + 2, sin(2t), 4r“) Find the indefinite vector integral F(t) dt = ( ] ) + (cı,C,cs) where (C, C2, C3 ) is a vector of arbitrary constants.
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