Calculate \(\frac{\partial(x,y)}{\partial(u,v)}\) if \(u=2x+y\) and \(v=x+\ln(2y)\).

Compute the flux of the vector field \({\bf F}=yz^2{\bf i}+x^2\cos z{\bf j}+xy{\bf k}\) over the paraboloid parametrized by \({\bf r}(u,v)=\langle u,v,1-u^2-v^2 \rangle,\quad u^2+v^2\leq 1\) with downward orientation. [HINT: Try integrating over a different surface]

Which of the following is the form of the particular solution of the differential equation \(y”-4y’+4y=e^{2t}\cos(t)\)as given by the method of undetermined coefficients?

Consider the triangle with vertices \(P=(1,0,1)\), \(Q=(3,0,0)\), and \(R=(2,-1,4)\). (i) Find a vector \(\mathbf{v}\) orthogonal to the triangle \(PQR\) (ii) Find the area of \(PQR\).

Find the domain of the vector function \({\bf r}(t)=\langle \ln(4-t^{2}), \sqrt{t+1} \rangle\)

Determine the length of the curve \({\bf r}(t) = \langle 2t, 3sin(2t), 3cos(2t) \rangle\) on the interval \(0 \leq t \leq \pi\).

What is the equation of the tangent plane to the surface given by the graph of the function \( g(x,y)=1-x^2-y^2\) at the point \((0,0,1)\)?

Find the flux of the vector field \({\bf F}= \bigg\langle \frac{-y}{x^2+y^2},\frac{x}{x^2+y^2},1 \bigg\rangle\) over the cylinder of radius 1, centered around the \(z\)-axis and between \(z=0\) and \(z=1\) with the disks at the top and bottom included so that the surface is closed.

Find the domain of the vector function \({\bf r}(t)=\langle \ln(4-t^{2}), \sqrt{t+1} \rangle\)

Let \(f(x,y,z)=xy+\cos z\) and \({\bf r} (t)=\langle t^2,t,\frac{\pi}{2}\sin^3(t+\frac{\pi}{2}) \rangle\) and let \(g(t)=f({\bf r} (t))\)   Compute \(g'(0).\)