Show that if u is a vector in R^{2} or R&3, then u+(−1)u=0 u+(−1)u=0

How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

Give a full correct answer for given question 1- Let W be the set of all polynomials ${\displaystyle a+bt+c{t}^2\in P_2}$ such that ${\displaystyle a+b+c=0}$ Show that W is a subspace of ${\displaystyle P_2,}$ find a basis for W, and then find dim(W) 2 - Find two different bases of ${\displaystyle R^2}$ so that the coordinates of $b=\left[\begin{array}{c}5\\ 3\end{array}\right]$ are both (2,1) in the coordinate system defined by these two bases

Given point P(-2, 6, 3) and vector ${\displaystyle B=y{a}_x+(x+z)a_y}$, express P and B in cylindrical and spherical coordinates. Evaluate A at P in the Cartesian, cylindrical and spherical systems.

To find: The alternate solution to the exercise with the help of Lagrange Multiplier. ${\displaystyle x+2y+3z=6}$

Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed
$a)\overrightarrow{A}(x,y,z)={\overrightarrow{e}}_x$
$d)\overrightarrow{A}(\rho ,\varphi ,z)={\overrightarrow{e}}_{\rho }$
$g)\overrightarrow{A}(r,\theta ,\varphi )={\overrightarrow{e}}_{\theta }$
$j)\overrightarrow{A}(x,y,z)=\frac{-y{\overrightarrow{e}}_x+x{\overrightarrow{e}}_y}{x^2+y^2}$

The quadratic function ${\displaystyle y=a{x}^2+bx+c}$ whose graph passes through the points (1, 4), (2, 1) and (3, 4).

The equivalent polar equation for the given rectangular - coordinate equation.
${\displaystyle y=-3}$