Advanced Math

Question : Parameterize the plane which contains the point (1, 1, 1) and which is parallel to the vectors (1, 0, −1) and (1, −1, 0). Solution : Let a,b,c…

Question : Construct a potential function for the vector field (e^z, 6, xe^z). Solution : Potential Function : A scalar function ϕ is called potential function for a vector field…

Question : Use Gauss Divergence theorem to evaluate the surface integral over the portion of z=9-x^2-y^2 with z>=0 \[\iint_{\gamma}{\left(\nabla\ \times\ \vec{F}\right).\hat{n}\ dS\ }\]where\[ \vec{F}\ =\ (y-z)\ \hat{i}-(x+z)\hat{j}\ +\ (x+y)\ \hat{k}…

Question : Use Gauss theorem to evaluate the surface integral\[\iint_{\gamma}{\left(x^2+y+z\right)dS\ }\]Where γ is the surface of the ball\[ x^2+y^2+z^2\le1\] Solution : Unit Normal vector to surface \[\gamma\ =x^2+y^2+z^2-1\]Is given as\[\hat{n}=\frac{\nabla\gamma}{\left|\nabla\gamma\right|}\]…