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Section 16.1 : Vector Fields

1. Sketch the vector field for \(\vec F = - {y^2}\,\vec i + x\,\vec j\).
2. Sketch the vector field for \(\vec F = \,\vec i + xy\,\vec j\).
3. Sketch the vector field for \(\vec F = \,4y\,\vec i + \left( {x + 2} \right)\vec j\).
4. Compute the gradient vector field for \(f\left( {x,y} \right) = 6{x^2} - 9y + {x^3}\,\sqrt y \).
5. Compute the gradient vector field for \(f\left( {x,y} \right) = \sin \left( {2x} \right)\cos \left( {3x} \right)\).
6. Compute the gradient vector field for \(f\left( {x,y,z} \right) = z\,{{\bf{e}}^{x\,y}} + {y^3}\tan \left( {4x} \right)\).
7. Compute the gradient vector field for \(f\left( {x,y,z} \right) = x\,{y^2}{z^3} + 4x{{\bf{e}}^{{y^{\,2}}}} - \ln \left( {x - z} \right)\).