S-1. Consider the functions f: RR defined by f(x, y, 2) 2-y2- and g(r,y,) -2 Describe the sets of...
Consider the surface defined by 2 = f(x,y), where f(x, y) = (x + y2 - 1)(x + y - 4). (a) In three separate diagrams draw the level sets of the function at C=2, C = 4, and C= 6. State the coordinates of any isolated points and the radii of any circles that make up these level sets. (Hint: To get an idea of what the surface looks like it might help to look at the curves f(0,y)...
(1 point) Consider the function defined by F(x, y) = x2 + y2 except at (r, y) - (0, 0) where F(0,0)0 Then we have (0,0) = (0,0) = ax dy Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at (0,0) Note: You can earn partial credit on this problem...
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction
Exercise 2 We consider the functional F: X → R defined by Find the minimum and al the minimizers of the functional F over the set X in the following cases X-(continuous functions y: 10,11 → R} b. X = {continuous functions y: 10, 1] → R such that y(0-1) Hint: use Exercise 1
2. Let f:R2 + R be defined by gry, if (x, y) + (0,0) f(x,y) := { x2 + y2 + 1 0 if (x, y) = (0,0). Show that OL (0, y) = 0 for all y E R and f(x,0) = x for all x E R. Prove that bebu (0,0) + (0,0).
Suppose that the functions q and r are defined as follows. g(x)=-x+2 r(x)=x²+1 Find the following. (y - z)(-2) = 0 (20") (-2) = 0 x 3 ?
(a). Consider the field F(xyz-1 (x + y2) і-1 (x2 + y) j + 1 (x + Y) k. Integrate F around the rectangle defined by the points (21,0), (3.5.1.0). (3.5.3.90), and (2.3.9,0). (b). Consider the field given in Part (a). Integrate F over the surfaces of a rectangular prism defined by the points given in Part (a) and extruded in the positive k direction by a distance 0.6 Give answers to three decimal places. Answer for Part (a): Answer...
Question 1. Consider these real-valued functions of two variables: T In (x2 + y2) (a) () What is the maximal domain, D, for the functions f and g? Write D in set notation. (ii) What is the range of f and g? Is either function onto? ii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z00, 2, 04 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...
Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions over AABC. Calculate the values of the degrees of freedom Ci in the linear combination that represents f(x,y): f(x, y)- CiN(x, y) T3 finite element is defined over ΔABC (in physical coordinates). The vertices of this triangle have the following coordinates: A(-2,-1), B(3,2), and C(0, 6) Problem 1 Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions...