4. (a) Verify the divergence theorem in the plane for Fryi+ (2x - y)j and C is the triangle with vertices at (0,0), (1,0) and (0, 1). (b) Find the circulation of F around the curve C above. (c) If (velocity), give physical interpretations of funds and § ¨· dÄ. =
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- Let vector r(t) = <5sin(t), 5cos(t), 2t^(3/2)> be the position function of an object.a. Find vector v(t) and vector a(t), the velocity and acceleration of the object.b. Find the speed of the object.c. Find the distance the object travels on the interval [0, 1]. Also find the average speed of theobject on this interval.B A' A is the point (3, 1). Bis the point (a, 11). The gradient of AB is 5/2 -1 2 O 7Determine the gradient of the curve 3xy + y? = -2 at the point (1, -2)
- Derive the relation A,(to) = 47 S, (r)², as given in Equation 6.24, starting from the Robertson-Walker metric of Equation 6.22. 6.7 ds? = -c²dr² + a(t)°[dr² + S« (r)²d$²), (6.22) Ap(to) = 4x S, (r)². (6.24)Which of the following functions are linear? Justify your answer in a comprehensible manner. In the case of a linear function, determine the basis of the image and the core and also the associated dimensions in a comprehensible manner. (i) fi: R R, (a, b,c)"() (ii) fa : R - R. (:) ) (ii) f; : R → R', (:) ) (iv) fa : R → R', (a, b,c) (1,1, 1)"show full and complete procedure HANDWRITTEN only
- 2. [-/16.66 Points] DETAILS Find the directional derivative of the function at the given point in the direction of the vector v. х, у, 2) %3Dхуz (4, 2, 8), v = -1, –2, 2] Duf(4, 2, 8) =Suppose you are climbing a hill whose shape is given by the equation z = 1000 −0.005x2 −0.01y2 where x, y,and z are measured in meters, and you are standing at a point with coordinates (60, 40, 966). The positivex−axis points east and the positive y−axis points north. Find a vector representing the direction of largest slope at point P. What is the rate ofascent in that direction?Find a vector representing the direction where there is zero slope at the point P .Suppose that the directional derivative of a function f at a point Pin the direction of the vector ū = (-1,0) is - Relate this 4 value to the behaviour of f. A. From the point P, the function f is increasing at a rate of - in the positive x-direction. B. From the point P, the function f is increasing at a rate of - in the positive y-direction. C. From the point P, the function f is decreasing at a rate of - in the negative y-direction.