Question 3 (a) Consider the bowl-shaped paraboloid, for s,. E R 0 If you are locasied on the paraboboid st ti (20th?). 2 4 direction should you move in order to ascend on the surface at the maximum r...
(9 points) Consider the bowl-shaped paraboloid z = f(x, y) = 4 + x2 + 3y2. (a) If you are located on the paraboloid at the point (2, - *), in which direction should you move in order to ascend on the surface at the maximum rate? What is the rate of change? (b) If you are located at the point (2, -), in which direction should you move in order to descend on the surface at the maximum rate?...
(4) Calculate the flux of the vector field F(x,y, 2)zr across the surface of the paraboloid ,; 1-2,2-y2 that lies above the x-y plane. Make sure that you specify the direction of positive flux. Show all working. (4) Calculate the flux of the vector field F(x,y, 2)zr across the surface of the paraboloid ,; 1-2,2-y2 that lies above the x-y plane. Make sure that you specify the direction of positive flux. Show all working.
Question 2. Consider a surface S in the 0 plane with three smooth boundary curves C1, C2, and C3 as shown in the diagram. Each curve is parametrised so that it is traversed in the direction shown by the arrows For a smooth vector field A(x, y, z) you are given the following results: Ca Adr =-3 C2 0.5 2.0 1.5 -0.5 (a) What is the value of the surface integral ▽ × A. ds. if we assume by convention...
PLEASE DO LETTER d.) PLEASE DO LETTER f.) The plane from e.) is 4(x-2)+6(y-1)+(z-1)=0 or 4x+6y+z=15 15. The temperature on an unevenly heated metal plate positioned in the first quadrant of the xy-plane is given by 25xy + 25 C(x, y) = 7 (x - 1)2 + (y - 1)2 +1° Assume that temperature is measured in degrees Celsius and that x and y are each measured in inches. (Note: At no point in the following questions should you expand...
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...