158 exercises 1 compute the partial derivatives of

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15.8 Exercises 1. Compute the partial derivatives of the functions f (x,y) = arcsin y x , g(x,y) = log 1 x 2 + y 2 . Verify your results with maple . 2. Show that the function v(x,t) = 1 t exp x 2 4 t satisfies the heat equation v t = 2 v x 2 for t > 0 and x R . 3. Show that the function w(x,t) = g(x kt) satisfies the transport equation w t + k w x = 0 for any differentiable function g .
210 15 Scalar-Valued Functions of Two Variables 4. Show that the function g(x,y) = log (x 2 + 2 y 2 ) satisfies the equation 2 g x 2 + 1 2 2 g y 2 = 0 for (x,y) ̸ = ( 0 , 0 ) . 5. Represent the ellipsoid x 2 + 2 y 2 + z 2 = 1 as graph of a function (x,y) f (x,y) . Distinguish between positive and negative z -coordinates, respectively. Compute the partial derivatives of f , and sketch the level curves of f . Find the direction in which f points. 6. Solve Exercise 5 for the hyperboloid x 2 + 2 y 2 z 2 = 1. 7. Consider the function f (x,y) = y e 2 x y , where x = x(t) and y = y(t) are dif- ferentiable functions satisfying x( 0 ) = 2 , y( 0 ) = 4 , ˙ x( 0 ) = 1 , ˙ y( 0 ) = 4 . From this information compute the derivative of z(t) = f (x(t),y(t)) at the point t = 0. 8. Find all stationary points of the function f (x,y) = x 3 3 xy 2 + 6 y. Determine whether they are maxima, minima or saddle points. 9. Investigate the function f (x,y) = x 4 3 x 2 y + y 3 for 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- pings f (x, 0 ) and f ( 0 ,y) . 10. Determine for the function f (x,y) = x 2 e y/ 3 (y 3 ) 1 2 y 2 (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 saddle points.
16 Vector-Valued Functions of Two Variables In this section we briefly touch upon the theory of vector-valued functions in several variables. 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 of continuity and differentiability to vector-valued functions. Then we discuss Newton’s method in two variables. As an application we compute a common zero of two nonlinear functions. Finally, as an extension of Sect. 15.1 , we show how smooth surfaces can be described mathematically with the help of parametrisations. For the required basic notions of vector and matrix algebra we refer to Appen- dices A and B . 16.1 Vector Fields and the Jacobian In the entire section D denotes an open subset of R 2 and F : D R 2 R 2 : (x,y) u v = F (x,y) = f (x,y) g(x,y) a vector-valued function of two variables with values in R 2 . Such functions are also called vector fields since they assign a vector to every point in the plane. Important applications are provided in physics. For example, the velocity field of a flowing liquid or the gravitational field are mathematically described as vector fields.

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