1.3 Integral Calculus (iv) 1 ZA 1 (v) (ii) (vi) FIGURE 1.29 1 So much for the left side of the divergence theorem. To evaluate the surface integral we must consider separately the six faces of the cube: (i) (ii) (iii) (iv) (v) (vi) So the total flux is: 1 1 [ v · da = f' f' y²dy dz = } [ v. da v.da [ v · da = L 1 1 - S² S² y² dy dz = − } . [ = Sv. $ S v.da L 1 1 (2x + z²) dx dz = 3. 1 - S' S² ² dx d² = − } . L 1 1 [ v-da = L'S' dz S' S² 2y dx dy = 1. Sv da = - S' S L L · ½ 1 0dx dy = 0. v da = -+-+1+0=2, as expected. 33 33 Example 1.10. Check the divergence theorem using the function v = y²â+ (2xy + z²) ŷ + (2yz) z and a unit cube at the origin (Fig. 1.29). Solution In this case V.v=2(x + y), and 1 {2(x + y) dt = 2 f f f (x + y)dx dy dz. 1 S' L' = (x + y) dx = + y, = 1+y, f² (4 + y) dy = 1, [ '1dz = 1. 0 Thus, V.vdt = 2.

1.3 Integral Calculus (iv) 1 ZA 1 (v) (ii) (vi) FIGURE 1.29 1 So much for the left side of the divergence theorem. To evaluate the surface integral we must consider separately the six faces of the cube: (i) (ii) (iii) (iv) (v) (vi) So the total flux is: 1 1 [ v · da = f' f' y²dy dz = } [ v. da v.da [ v · da = L 1 1 - S² S² y² dy dz = − } . [ = Sv. $ S v.da L 1 1 (2x + z²) dx dz = 3. 1 - S' S² ² dx d² = − } . L 1 1 [ v-da = L'S' dz S' S² 2y dx dy = 1. Sv da = - S' S L L · ½ 1 0dx dy = 0. v da = -+-+1+0=2, as expected. 33 33 Example 1.10. Check the divergence theorem using the function v = y²â+ (2xy + z²) ŷ + (2yz) z and a unit cube at the origin (Fig. 1.29). Solution In this case V.v=2(x + y), and 1 {2(x + y) dt = 2 f f f (x + y)dx dy dz. 1 S' L' = (x + y) dx = + y, = 1+y, f² (4 + y) dy = 1, [ '1dz = 1. 0 Thus, V.vdt = 2.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

Figure 1.30
Do the problem in 1st quandrant. Explain how you determined the sign of each derivative. Draw pictures to illustrate your reasoning.
1.32 In the Eigen vector equation AX = X, the operator A is given 32 A = 41 Find: (a) The Eigen values (b) The Eigen vector X (c) The modal matrix C and it's inverse C-1 (d) The product C-1 AC
1 W:0E *Problem 1.3 Consider the gaussian distribution p(x) = Ae¬^(x-a)² %3D where A, a, and A are positive real constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. (b) Find (x), (x²), and ơ. (c) Sketch the graph of p(x).
Subject: Mathematical Physics Topic: Functions of A Complex Variable-Analytic Function Answer the following functions and determine whether the following functions are Analytic. Please answer neatly and with details.
Subject: Mathematical Physics Topic: Functions of A Complex Variable-Analytic Function Answer the following functions and determine whether the following functions are Analytic. Please answer neatly and with details.
U sing the rules of B olo gnese algebra Demoreken's theorem, s implify the following. Booleah ex pression to the sim plest form and then draw the logical circle before simplification a fter simpiification an d the truth table (Ā + ē ).(B+ Ĉ ).(A+B +é)
Subject: Mathematical Physics Topic: Functions of A Complex Variable-Analytic Function Answer the following functions and determine whether the following functions are Analytic. Please answer neatly and with details.
1.3. Determine an orthogonal basis for the subspace of C (-1, 1] spanned by functions: {f(x) = x, f(x) = x³, f(x) = x³] using Gram-Schmidt process.
Problems 5.1 Determine the material derivative of the flux of any vector property Qj through the spatial area S. Specifically, show that in agreement with Eq 5.2-5. 1999 by CRC Press LLC 5.2 Let the property P in Eq 5.2-1 be the scalar 1 so that the integral in that equation represents the instantaneous volume V. Show that in this case dV = 5.3 Verify the identity !! and, by using this identity as well as the result of Problem 5.1, prove that the material derivative of the vorticity flux equals one half the flux of the curl of the acceleration; that is, show that 5.4 Making use of the divergence theorem of Gauss together with the identity aw at show that w,v, - dt
Evaluate the integral as either a volume integral or a surfaceintegral, whichever is easier. Please provide any necessary diagrams to further explain steps to the soulstion.
Problem 2.24 Delta functions live under integral signs, and two expressions (D₁ (x) and D₂(x)) involving delta functions are said to be equal if [ + f(x)D₁(x) dx = [+0° f(x)D₂(x) dx. -8 -00 for every (ordinary) function f(x). (a) Show that 8(cx) = -178(x). [2.142] where c is a real constant. (Be sure to check the case where c is negative.) (b) Let 0(x) be the step function: { 0(x)= 1. [2.143] (In the rare case where it actually matters, we define (0) to be 1/2.) Show that de/dx = 8(x). if x > 0. if x < 0. 0.