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15.8Exercises1. Compute the partial derivatives of the functionsf (x,y)=arcsinyx,g(x,y)=log1x2+y2.Verify your results withmaple.2. Show that the functionv(x,t)=1√texp−x24tsatisfies theheat equation∂v∂t=∂2v∂x2fort >0 andx∈R.3. Show that the functionw(x,t)=g(x−kt)satisfies thetransport equation∂w∂t+k∂w∂x=0for any differentiable functiong.
21015Scalar-Valued Functions of Two Variables4. Show that the functiong(x,y)=log(x2+2y2)satisfies the equation∂2g∂x2+12∂2g∂y2=0for(x,y)̸=(0,0).5. Represent the ellipsoidx2+2y2+z2=1 as graph of a function(x,y)→f (x,y). Distinguish between positive and negativez-coordinates, respectively.Compute the partial derivatives off, and sketch the level curves off. Find thedirection in which∇fpoints.6. Solve Exercise 5 for the hyperboloidx2+2y2−z2=1.7. Consider the functionf (x,y)=ye2x−y, wherex=x(t)andy=y(t)are dif-ferentiable functions satisfyingx(0)=2,y(0)=4,˙x(0)=−1,˙y(0)=4.From this information compute the derivative ofz(t)=f (x(t),y(t))at thepointt=0.8. Find all stationary points of the functionf (x,y)=x3−3xy2+6y.Determine whether they are maxima, minima or saddle points.9. Investigate the functionf (x,y)=x4−3x2y+y3for local extrema and saddle points. Visualise the graph of the function.Hint. To study the behaviour of the function at(0,0)consider the partial map-pingsf (x,0)andf (0,y).10. Determine for the functionf (x,y)=x2ey/3(y−3)−12y2(a) the gradient and the Hessian matrix(b) the second-order Taylor approximation at(0,0)(c) all stationary points. Find out whether they are maxima, minima or saddlepoints.
16Vector-Valued Functions of Two VariablesIn this section we briefly touch upon the theory of vector-valued functions in severalvariables. To simplify matters we limit ourselves again to the case of two variables.First, we define vector fields in the plane and extend the notions ofcontinuityanddifferentiabilityto vector-valued functions. Then we discuss Newton’s methodin two variables. As an application we compute a common zero of two nonlinearfunctions. Finally, as an extension of Sect.15.1, we show how smooth surfaces canbe described mathematically with the help of parametrisations.For the required basic notions of vector and matrix algebra we refer to Appen-dicesAandB.16.1Vector Fields and the JacobianIn the entire sectionDdenotes an open subset ofR2andF:D⊂R2→R2:(x,y)→uv=F(x,y)=f (x,y)g(x,y)avector-valuedfunction of two variables with values inR2. Such functions are alsocalledvector fieldssince they assign a vector to every point in the plane. Importantapplications are provided in physics. For example, the velocity field of a flowingliquid or the gravitational field are mathematically described as vector fields.