2. Suppose that U(x, y, z) = x² + y²+ z² represents the temperature of a 3-dimensional solid objectat any point (x, y, z). ThenF(x, y, z) = -KVU (x, y, z)represents the heat flow at (x, y, z) where K > 0 is called the conductivity constant and thenegative sign indicates that the heat moves from higher temperature region into lower temperatureregion. Answer the following questions.(A) [90%] Compute the inward heat flux (i.e., the inward flux of F) across the surface z =1 - x² - y².(B) [10%] Use the differential operator(s) to determine if the heat flow is rotational or irrotational.

Approach to solving the question in an Easy Way: We are given the temperature…

Answered: 2. Suppose that U(x, y, z) = x² + y²+…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Use chain rule to find dz/dt.
2). Find the flux of F(x, y, z) = (0, yz, z) outward through the surface S cut from the cylinder y + z? = 1, z > 0 by the planes x = 0 and x = 1.
Find the flux of F = 3zi + 5y2 j+ 3x k out of the closed cone y = Vx2 + z2, with 0 < y<1. flux =
2) Find the outward flux of F across the region D where F(x, y, z) = (In(x² + y²), –tan-(), z Vx² + y² ) and the region D is the thick-walled cylinder 1 < x + y < 2 and –1 < z< 2.
az. Suppose F = (2xz + 3y²) a, + (4yz²) a;. (a) Calculate S[F·dS, where S is the shaded surface in Figure 1. (c) Based on your results for parts (a) and (b), what named theorem do you think is being satisfied here, if any? (b) Calculate SF· dl, where C is the A → B → C → D → A closed path in Figure 1. az C C (0,1,1) D (0,0,0) (A ay В ax Figure 1: Figure for Problem 1.
2. Let closed curve C consist of the line segment from (0,0) to (2, 1), followed by the line segment from (2, 1) to (-1,1), followed by the piece of y = x² from (-1,1) to (0,0). Also, let F = (-1,2y?). a) Find F-Ťds F-Ť ds by integrating F over each piece of C and adding the results. b) Find F.Ť ds by first converting it to a double integral using Greene's Theorem.
Find the flux F(x, y, z) = xi + 2yj +4zk, S is the cube with vertices (1, 1, 1), (-1, -1, -1)
a) Find the total differential for the following function z = f(x,y) = xy + y xb) Use Chain Rule for find, and if z x = st, at For the function x² + 2y²+3z² = 6, find and Əz 3x az y= s²-t².
Find the mass M of a fluid with a constant mass density flowing across the paraboloid z = 36 - x² - y², z ≥ 0, in a unit of time in the direction of the outer unit normal if the velocity of the fluid at any point on the paraboloid is F = F(x, y, z) = xi + yj + 13k. (Express numbers in exact form. Use symbolic notation and fractions where needed.) M = Incorrect 31740πσ > Feedback Use A Surface Integral Expressed as a Double Integral Theorem. Let S be a surface defined on a closed bounded region R with a smooth parametrization r(r, 0) = x(r, 0) i + y(r, 0) j + z(r, 0) k. Also, suppose that F is continuous on a solid containing the surface S. Then, the surface integral of F over S is given by [[F(x, y, z) ds = = D₁² Recall that the mass of fluid flowing across the surface S in a unit of time in the direction of the unit normal n is the flux of the velocity of the fluid F across S. M = F(x(r, 0), y(r, 0), z(r, 0))||1r × 1e|| dr de F.ndS X
8. Find the flow of the velocity field F(r, y) = (x + y)i – (x² + y²)j along the circle x² + y² = 1 from (1,0) to (0, 1).
Let w=−2xy−5yz−7xz,x=st,y=e^(st),z=t^2 Compute∂w/∂s(1,−2)=∂w/∂t(1,−2)=
fast please

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