To find: The equivalent polar equation for the given rectangular-coordinate equation.Given:\displaystyle{x}^{2}+{y}^{2}+{8}{x}={0}

How do you determine if the vectors are coplanar: a = [-2,-1,4], b = [5,-2,5], and c = [3,0,-1]?

I have a non-linear form of the Poisson equation (with a diffusion coefficient that is a function of the derivatives of the dependent variable) that I'm trying to solve numerically (using FEM which requires the pde to be posed in its weak form).
$\mathrm{\nabla }\cdot (\eta \mathrm{\nabla }v)=G(x)$
where $\eta =\sqrt{1/2(\mathrm{\nabla }v:\mathrm{\nabla }v)}$ and $v=v(y,z)$. Multiplying by a test function θ and integrating wrt the domain, $\mathrm{\Omega }$, gives the weak form
${\oint }_{\mathrm{\Gamma }}\theta \cdot (\eta \mathrm{\nabla }v\cdot \mathbf{n})d\mathrm{\Gamma }-{\int }_{\mathrm{\Omega }}(\eta \mathrm{\nabla }v\cdot \mathrm{\nabla }\theta +G)d\mathrm{\Omega }=0$
However, none of my boundary conditions correspond to the Dirichelet type as I have ${\mathrm{\partial }}_{n}v=v$ on three of the boundaries in a rectangular domain and ${\mathrm{\partial }}_{n}v=0$ on the other (n is the normal vector to surface). Is this problem actually solvable using FEM, or at all? I have the solution to a simpler problem which might be a reasonable guess at the solution here.

Find the equation of the line that satisfies the given conditions:
Goes through $(-1,-2)$ and is parallel to the line $2x+3y=8$

If the Wro
ian W of f and g is ${\displaystyle 3e^{4t}}$, and if ${\displaystyle \left(t\right)=e^{2t}}$, find g(t).

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. $\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 0& 3\\ 0& 0& 1& 4\\ 0& 0& 0& 0\end{array}\right]$

Let there be a linear transformation going from ${\mathbb{R}}^3$ to ${\mathbb{R}}^2$, defined by $T(x,y,z)=(x+y,2z-x)$. Find the transformation matrix if base 1:
$⟨(1,0,-1),(0,1,1),(1,0,0)⟩$,
base 2: $⟨(0,1),(1,1)⟩$
An attempt at a solution included calculating the transformation on each of the bases in ${\mathbb{R}}^3$, (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of $B_1$ in $B_2$
Another point: if the basis for ${\mathbb{R}}^3$ and ${\mathbb{R}}^2$ are the standard basis for these spaces, the attempt at a solution is a correct answer.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.