Multivariable Calculus

Question : If u=sin^(-1){(x+2y+3z)/(x^8+y^8+z^8)} then find the value of xdf/dx+ydf/dy+zdf/dz Solution : If \[u=sin^{-1}\left(\frac{x+2y+3z}{x^8+y^8+z^8}\right)\] then=>\[ sin{\left(u\right)}=\frac{x+2y+3z}{x^8+y^8+z^8}\]Differentiate with respect to x\[cos{\left(u\right)}\frac{\partial u}{\partial x}=\frac{\left(x^8+y^8+z^8\right).1-\left(x+2y+3z\right).\left(8x^7\right)}{\left(x^8+y^8+z^8\right)^2}……….(1)\]Similarly differentiating with respect to y and z \[cos{\left(u\right)}\frac{\partial…

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\[\hat{n}=\frac{\nabla\gamma}{\left|\nabla\gamma\right|}\]…

How to find roots of x^3-3x^2-16x+48=0 if the sum of two of its roots being equal to zero ? Answer : To find roots of x^3-3x^2-16x+48=0 if the sum of…

Question : Write a matlab program to solve x+y=3 y+z=5 z+x=4 by cramer rule Solution: % Cramer Rule to solve system of linear equations, AX = B;% Let's denote matrices…