Examples of divergence theorem

Divergence Theorem is a theorem that talks about the flux of a vector field through a closed area to the volume enclosed in the divergence of the field. ... gravity, and examples in quantum physics like probability density. It can also be applied in the aerodynamic continuity equation-

7.3. EXTENSION TO GAUSS’ THEOREM 7/5 Thisisstillascalarequationbutwenownotethatthevectorc isarbitrarysothatthe resultmustbetrueforanyvectorc. ThiscaMultivariable calculus 5 units · 48 skills. Unit 1 Thinking about multivariable functions. Unit 2 Derivatives of multivariable functions. Unit 3 Applications of multivariable derivatives. Unit 4 Integrating multivariable functions. Unit 5 Green's, Stokes', and the divergence theorems.In mathematics, the divergence theorem is a theorem about vector fields. It states that the divergence of a vector field is zero in a region if and only if the field is the gradient of a scalar field. The theorem is named for the mathematician George Green, who stated it in 1828. The theorem is also known as the Kelvin-Stokes theorem, after ...View Answer. Use the Divergence Theorem to calculate the surface integral \iint F. ds; that is calculate the flux of F across S: F (x, y, z) = xi - x^2j + 4xyzk, S is the surface of the solid bounded by the cyl... View Answer. Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.How do you use the divergence theorem to compute flux surface integrals?

Open this example in Overleaf. This example produces the following output: The command \theoremstyle { } sets the styling for the numbered environment defined right below it. In the example above the styles remark and definition are used. Notice that the remark is now in italics and the text in the environment uses normal (Roman) typeface, the ...This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix) is the gradient of a vector function . = (, ) =, = (), = [()] = (, ) =, = = The last equation is ... When is equal to the identity tensor, we get the divergence theorem =. We can express the formula for integration by parts in Cartesian index ...(3) Verify Gauss' Divergence Theorem. In these types of questions you will be given a region B and a vector field F. The question is asking you to compute the integrals on both sides of equation (3.1) and show that they are equal. 4. EXAMPLES Example 1: Use the divergence theorem to calculate RR S F·dS, where S is the surface ofExplore Stokes' theorm and divergence theorem - example 1 explainer video from Calculus 3 on Numerade.View Answer. Use the Divergence Theorem to calculate the surface integral \iint F. ds; that is calculate the flux of F across S: F (x, y, z) = xi - x^2j + 4xyzk, S is the surface of the solid bounded by the cyl... View Answer. Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.Bringing the boundary to the interior. Green's theorem is all about taking this idea of fluid rotation around the boundary of R , and relating it to what goes on inside R . Conceptually, this will involve chopping up R into many small pieces. In formulas, the end result will be taking the double integral of 2d-curl F .

The divergence theorem lets you translate between surface integrals and triple integrals, but this is only useful if one of them is simpler than the other. In each of the following examples, take note of the fact that the volume of the relevant region is simpler to describe than the surface of that region.So hopefully this gives you an intuition of what the divergence theorem is actually saying something very, very, very, very-- almost common sense or intuitive. And now in the next few videos, we can do some worked examples, just so you feel comfortable computing or manipulating these integrals.(c) Gauss’ theorem that relates the surface integral of a closed surface in space to a triple integral over the region enclosed by this surface. All these formulas can be uni ed into a single one called the divergence theorem in terms of di erential forms. 4.1 Green’s Theorem Recall that the fundamental theorem of calculus states that b aboundary, the volume of a region can be computed as a flux integral: Take for example the vector field F(x, y, z) = 〈x, 0, 0〉 which has divergence 1. The flux ...divergence theorem is done as in three dimensions. By the way: Gauss theorem in two dimensions is just a version of Green's theorem. Replacing F = (P;Q) with G= ( Q;P) gives curl(F) = div(G) and the ux of Gthrough a curve is the lineintegral of Falong the curve. Green's theorem for Fis identical to the 2D-divergence theorem for G.Yep. 2z, and then minus z squared over 2. You take the derivative, you get negative z. Take the derivative here, you just get 2. So that's right. So this is going to be equal to 2x-- let me do that same color-- it's going to be equal to 2x times-- let me get this right, let me go into that pink color-- 2x times 2z.

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Open this example in Overleaf. This example produces the following output: The command \theoremstyle { } sets the styling for the numbered environment defined right below it. In the example above the styles remark and definition are used. Notice that the remark is now in italics and the text in the environment uses normal (Roman) typeface, the ...Example Verify the Divergence Theorem for the region given by x2 + y2 + z2 4, z 0, and for the vector eld F = hy;x;1 + zi. Computing the surface integral The boundary of Wconsists of the upper hemisphere of radius 2 and the disk of radius 2 in the xy-plane. The upper hemisphere is parametrized byIn this section and the remaining sections of this chapter, we show many more examples of such series. Consequently, although we can use the divergence test to show that a series diverges, we cannot use it to prove that a series converges. Specifically, if \( a_n→0\), the divergence test is inconclusive.In this theorem note that the surface S S can actually be any surface so long as its boundary curve is given by C C. This is something that can be used to our advantage to simplify the surface integral on occasion. Let’s take a look at a couple of examples. Example 1 Use Stokes’ Theorem to evaluate ∬ S curl →F ⋅ d →S ∬ S curl F ...How do you use the divergence theorem to compute flux surface integrals?

This integral is called "flux of F across a surface ∂S ". F can be any vector field, not necessarily a velocity field. Gauss's Divergence Theorem tells us that ...Figure 1: The divergence theorem relates a flux integral across a closed surface S to a triple integral over solid E enclosed by the surface. Recall that the flux form of Green's theorem states that. ∬Ddiv ⇀ FdA = ∫C ⇀ F ⋅ ⇀ NdS. Therefore, the divergence theorem is a version of Green's theorem in one higher dimension.The Gauss divergence theorem, which serves as the foundation of the finite volume method, is first ascribed a physical interpretation. ... Consider, for example, the convective fluxes in the x direction. One determines in general the value of a variable (e.g. pressure or velocity) at the location x by employing an interpolation polynomial ...In this video section I derive the Divergence Theorem.This video is part of a Complex Analysis series where I derive the Planck Integral which is required in...Multivariable Taylor polynomial example. Introduction to local extrema of functions of two variables. Two variable local extrema examples. Integral calculus. Double integrals. Introduction to double integrals. Double integrals as iterated integrals. Double integral examples. Double integrals as volume.The Divergence. The divergence of a vector field. in rectangular coordinates is defined as the scalar product of the del operator and the function. The divergence is a scalar function of a vector field. The divergence theorem is an important mathematical tool in electricity and magnetism.Aug 20, 2023 · The divergence theorem is a higher dimensional version of the flux form of Green’s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus. The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa. Example 4.1.2. As an example of an application in which both the divergence and curl appear, we have Maxwell's equations 3 4 5, which form the foundation of classical electromagnetism.

Gauss's Divergence theorem is one of the most powerful tools in all of mathematical physics. It is the primary building block of how we derive conservation ...

Stokes' Theorem and Divergence Theorem Problem 1 (Stewart, Example 16.8.1). Find the line integral of the vector eld F= h y 2;x;ziover the curve Cof intersection of the plane x+ z= 2 and the cylinder x 2+ y = 1 knowing that C is oriented counterclockwise when viewed from above. [Answer: ˇ] Problem 2 (Stewart, Example16.8.1).We give a verification example involving the divergence theorem.Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1Personal Website:...When you learn about the divergence theorem, you will discover that the divergence of a vector field and the flow out of spheres are closely related. For a basic understanding of divergence, it's enough to see that if a fluid is expanding (i.e., the flow has positive divergence everywhere inside the sphere), the net flow out of a sphere will be positive. …Price divergence is unrealistic and not empirically seen. The idea that farmers only base supply on last year’s price means, in theory, prices could increasingly diverge, but farmers would learn from this and pre-empt …The Divergence Theorem In this section, we will learn about: The Divergence Theorem for simple solid regions, and its applications in electric fields and fluid flow. 4 . INTRODUCTION • In Section 16.5, we rewrote Green’s Theorem in a vector version as: • where C is the positively oriented boundary curve of the plane region D. div ( , ) C ...This video lecture of Vector Calculus - Gauss Divergence Theorem | Example and Solution by vijay sir will help Bsc and Enginnering students to understand fo...Yes, the normal vector on a cylinder would be just as you guessed. It's completely analogous to z^ z ^ being the normal vector to a surface of contant z z, such as the xy x y -plane or any plane parallel to it. David H about 9 years. Also, your result 6 3-√ πa2 6 3 π a 2 is correct. Your calculation using the divergence theorem is wrong.The limit in this test will often be written as, c = lim n→∞an ⋅ 1 bn c = lim n → ∞ a n ⋅ 1 b n. since often both terms will be fractions and this will make the limit easier to deal with. Let's see how this test works. Example 4 Determine if the following series converges or diverges. ∞ ∑ n=0 1 3n −n ∑ n = 0 ∞ 1 3 n − n.Verification of the Divergence Theorem Evaluate I (Ixi — ak) + nA over the sphere S: x +? + 2 =4 (a) by (2), (b) directly. Solution. (a) div F = iv (7.0. —2} () We can represent S by (3), See. 105 ( 'Accordingly, iv Uni — ck] = 7 — 1 = 6, Answer: 6 (dyer «2° = 64a. ih a = 2), and we shall use nd = N du do [see (3°), See. 1066], S: r= [Deosveosu, 2eoswsinu, 2sinu] Then j-2eosv sin ...This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Example Evaluate both sides of the divergence theorem for the field D = 2xy ax + x a, C/m2 and the rectangular parallelepiped formed by the planes x = 0 and 3, y = 0 and 1, and z= 0 and 2. V.

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Example I Example Verify the Divergence Theorem for the region given by x2 + y2 + z2 4, z 0, and for the vector eld F = hy;x;1 + zi. Computing the surface integral The boundary of Wconsists of the upper hemisphere of radius 2 and the disk of radius 2 in the xy-plane. The upper hemisphere is parametrized by Price divergence is unrealistic and not empirically seen. The idea that farmers only base supply on last year’s price means, in theory, prices could increasingly diverge, but farmers would learn from this and pre-empt …Example 2. For F = (xy2, yz2,x2z) F = ( x y 2, y z 2, x 2 z), use the divergence theorem to evaluate. ∬SF ⋅ dS ∬ S F ⋅ d S. where S S is the sphere of radius 3 centered at origin. Orient the surface with the outward pointing normal vector. Solution: Since I am given a surface integral (over a closed surface) and told to use the ... Definition 4.3.1 4.3. 1. A sequence of real numbers (sn)∞n=1 ( s n) n = 1 ∞ diverges if it does not converge to any a ∈ R a ∈ R. It may seem unnecessarily pedantic of us to insist on formally stating such an obvious definition. After all “converge” and “diverge” are opposites in ordinary English.The Monotone Convergence Theorem asserts the convergence of a sequence without knowing what the limit is! There are some instances, depending on how the monotone sequence is de ned, that we can get the limit after we use the Monotone Convergence Theorem. Example. Recall the sequence (x n) de ned inductively by x 1 = 1; x n+1 = (1=2)x n + 1;n2N:If lim n→∞an = 0 lim n → ∞ a n = 0 the series may actually diverge! Consider the following two series. ∞ ∑ n=1 1 n ∞ ∑ n=1 1 n2 ∑ n = 1 ∞ 1 n ∑ n = 1 ∞ 1 n 2. In both cases the series terms are zero in the limit as n n goes to infinity, yet only the second series converges. The first series diverges.An integral having either an infinite limit of integration or an unbounded integrand is called an improper integral. Two examples are. ∫∞ 0 dx 1 + x2 and ∫1 0dx x. The first has an infinite domain of integration and the integrand of the second tends to ∞ as x approaches the left end of the domain of integration.2. Stokes' Theorem and the Divergence Theorem both generalize two sides of Green's Theorem which was about a region in the 2D plane with a boundary. However, they generalize in different ways. Stokes' theorem is still comparing a surface integral to a line integral along the boundary, it is just the surface lives in 3D not 2D.Using the divergence theorem, the surface integral of a vector field F=xi-yj-zk on a circle is evaluated to be -4/3 pi R^3. 8. The partial derivative of 3x^2 with respect to x is equal to 6x. 9. A ...Divergence Theorem Theorem Let D be a nice region in 3-space with nice boundary S oriented outward. Let F be a nice vector field. Then Z Z S (F n)dS = Z Z Z D div(F)dV where n is the unit normal vector to S. Example Find the flux of F = xyi+yzj+xzk outward through the surface of the cube cut from the first octant by the planes x = 1, y = 1 ...If we think of divergence as a derivative of sorts, then the divergence theorem relates a triple integral of derivative divF over a solid to a flux integral of F over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field F over a closed surface S to a triple integral of the divergence of F ... This video explains how to apply the divergence theorem to determine the flux of a vector field.http://mathispower4u.wordpress.com/ ….

example, if volume V is a sphere, then S is the surface of that sphere. ... field! 9/16/2005 The Divergence Theorem.doc 2/2 Jim Stiles The Univ. of Kansas Dept. of EECS -4-20-4-2 0 2 4 What the divergence theorem indicates is that the total "divergence" of a vector field through the surface of any volume is equal to the sum (i.e ...15.7 The Divergence Theorem and Stokes' Theorem; Appendices; 15 Vector Analysis 15.1 Introduction to Line Integrals 15.3 Line Integrals over Vector Fields. 15.2 Vector Fields. ... One may find this curl to be harder to determine visually than previous examples. One might note that any arrow that induces a clockwise spin on a cork will have an ...In this example we use the divergence theorem to compute the flux of a vector field across the unit cube. Instead of computing six surface integral, the dive...The surface integral of f over Σ is. ∬ Σ f ⋅ dσ = ∬ Σ f ⋅ ndσ, where, at any point on Σ, n is the outward unit normal vector to Σ. Note in the above definition that the dot product inside the integral on the right is …The Divergence. The divergence of a vector field. in rectangular coordinates is defined as the scalar product of the del operator and the function. The divergence is a scalar function of a vector field. The divergence theorem is an important mathematical tool in electricity and magnetism.A Useful Theorem; The Divergence Test; A Divergence Test Flowchart; Simple Divergence Test Example; Divergence Test With Square Roots; Divergence Test with arctan; Video Examples for the Divergence Test; Final Thoughts on the Divergence Test; The Integral Test. A Motivating Problem for The Integral Test; A Second Motivating Problem for The ...The Pythagorean Theorem is the foundation that makes construction, aviation and GPS possible. HowStuffWorks gets to know Pythagoras and his theorem. Advertisement OK, time for a pop quiz. You've got a right-angled triangle — that is, one wh...My attempt at the question involved me using the divergence theorem as follows: ∬ S F ⋅ dS =∭ D div(F )dV ∬ S F → ⋅ d S → = ∭ D div ( F →) d V. By integrating using spherical coordinates it seems to suggest the answer is −2 3πR2 − 2 3 π R 2. We would expect the same for the LHS. My calculation for the flat section of the ...24.3. The theorem explains what divergence means. If we integrate the divergence over a small cube, it is equal the flux of the field through the boundary of the cube. If this is positive, then more field exits the cube than entering the cube. There is field “generated” inside. The divergence measures the “expansion” of the field ... Examples of divergence theorem, Stokes' theorem. Google Classroom. Assume that S is an outwardly oriented, piecewise-smooth surface with a piecewise-smooth, simple, closed boundary curve C oriented positively with respect to the orientation of S . ∮ C ( 4 y ı ^ + z cos ( x) ȷ ^ − y k ^) ⋅ d r. Use Stokes' theorem to rewrite the line integral as a surface integral., Jan 16, 2023 · The surface integral of f over Σ is. ∬ Σ f ⋅ dσ = ∬ Σ f ⋅ ndσ, where, at any point on Σ, n is the outward unit normal vector to Σ. Note in the above definition that the dot product inside the integral on the right is a real-valued function, and hence we can use Definition 4.3 to evaluate the integral. Example 4.4.1. , I'm confused about applying the Divergence theorem to hemispheres. Here is the statement: ... Divergence theorem is not working for this example? 2. multivariable calculus divergence theorem help. 0. Flux of a vector field across the upper unit hemisphere. Hot Network Questions, This video explains how to apply the divergence theorem to determine the flux of a vector field.http://mathispower4u.wordpress.com/, Apr 25, 2020 at 4:28. 1. Yes, divergence is what matters the sink-like or source-like character of the field lines around a given point, and it is just 1 number for a point, less information than a vector field, so there are many vector fields that have the divergence equal to zero everywhere. - Luboš Motl., Previous videos on Vector Calculus - https://bit.ly/3TjhWEKThis video lecture on 'Gauss Divergence Theorem | Vector Integration'. This is helpful for the st..., Oct 20, 2023 · The divergence theorem is the one in which the surface integral is related to the volume integral. More precisely, the Divergence theorem relates the flux through the closed surface of a vector field to the divergence in the enclosed volume of the field. It states that the outward flux through a closed surface is equal to the integral volume ... , In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.In particular, the …, This forms Gauss’ Theorem, or the Divergence Theorem. It states that the surface ... For example, consider a constant electric field: Ex=E0 ˆ . It is easy to see that the divergence of E will be zero, so the charge density ρ=0 everywhere. Thus, the total enclosed charge in any volume is zero, and by the integral form of Gauss’ Law the total flux through the surface …, Some examples of the 4-gradient as used in the d'Alembertian follow: ... More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a …, The Divergence Theorem in space Example Verify the Divergence Theorem for the field F = hx,y,zi over the sphere x2 + y2 + z2 = R2. Solution: Recall: ZZ S F · n dσ = ZZZ V (∇· F) dV. We start with the flux integral across S. The surface S is the level surface f = 0 of the function f (x,y,z) = x2 + y2 + z2 − R2. Its outward unit normal ... , Figure 16.7.1: Stokes’ theorem relates the flux integral over the surface to a line integral around the boundary of the surface. Note that the orientation of the curve is positive. Suppose surface S is a flat region in the xy -plane with upward orientation. Then the unit normal vector is ⇀ k and surface integral., Green's Theorem, Stokes' Theorem, and the Divergence Theorem 343 Example 1: Evaluate 4 C ∫x dx xydy+ where C is the positively oriented triangle defined by the line segments connecting (0,0) to (1,0), (1,0) to (0,1), and (0,1) to (0,0). Solution: By changing the line integral along C into a double integral over R, the problem is immensely simplified., No headers. The Divergence Theorem relates an integral over a volume to an integral over the surface bounding that volume. This is useful in a number of situations that arise in electromagnetic analysis. In this section, we derive this theorem. Consider a vector field \({\bf A}\) representing a flux density, such as the electric flux density \({\bf D}\) or magnetic flux density \({\bf B}\)., In this example we use the divergence theorem to compute the flux of a vector field across the unit cube. Instead of computing six surface integral, the dive..., Stokes' theorem is the 3D version of Green's theorem. It relates the surface integral of the curl of a vector field with the line integral of that same vector field around the boundary of the surface: ∬ S ⏟ S is a surface in 3D ( curl F ⋅ n ^) d Σ ⏞ Surface integral of a curl vector …, Level up on all the skills in this unit and collect up to 600 Mastery points! Here we cover four different ways to extend the fundamental theorem of calculus to multiple dimensions. Green's theorem and the 2D divergence theorem do this for two dimensions, then we crank it up to three dimensions with Stokes' theorem and the (3D) divergence theorem., The Gauss divergence theorem, which serves as the foundation of the finite volume method, is first ascribed a physical interpretation. ... Consider, for example, the convective fluxes in the x direction. One determines in general the value of a variable (e.g. pressure or velocity) at the location x by employing an interpolation polynomial ..., The 2D divergence theorem is to divergence what Green's theorem is to curl. It relates the divergence of a vector field within a region to the flux of that vector field through the boundary of the region. Setup: F ( x, y) ‍. is a two-dimensional vector field. R. ‍. is some region in the x y. , generalisations of the fundamental theorem of calculus to these vector spaces. These ideas provide the foundation for many subsequent developments in mathematics, most notably in geometry. They also underlie every law of physics. Examples of Maps To highlight some of the possible applications, here are a few examples of maps (0.1), Green's Theorem. Let C C be a positively oriented, piecewise smooth, simple, closed curve and let D D be the region enclosed by the curve. If P P and Q Q have continuous first order partial derivatives on D D then, ∫ C P dx +Qdy =∬ D ( ∂Q ∂x − ∂P ∂y) dA ∫ C P d x + Q d y = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y) d A. Before ..., Example 2. Verify the Divergence Theorem for F = x2 i+ y2j+ z2 k and the region bounded by the cylinder x2 +z2 = 1 and the planes z = 1, z = 1. Answer. We need to check (by calculating both sides) that ZZZ D div(F)dV = ZZ S F ndS; where n = unit outward normal, and S is the complete surface surrounding D. In our case, S consists of three parts ..., The Divergence Theorem in space Example Verify the Divergence Theorem for the field F = hx,y,zi over the sphere x2 + y2 + z2 = R2. Solution: Recall: ZZ S F · n dσ = ZZZ V (∇· F) dV. We start with the flux integral across S. The surface S is the level surface f = 0 of the function f (x,y,z) = x2 + y2 + z2 − R2. Its outward unit normal ..., The divergence theorem can also be used to evaluate triple integrals by turning them into surface integrals. This depends on finding a vector field whose divergence is equal to the given function. EXAMPLE 4 Find a vector field F whose divergence is the given function 0 aBb. (a) 0 aBb "SOLUTION (c) 0 aBb B# D # (b) 0 aBb B# C. The formula for ... , r= 1, the divergence test shows us the series diverges. Therefore the series converges exactly when jrj<1. With that assumption, taking the limit we have that S= lim n!1 S n= a 1 r (1 0) = a 1 r Examples Determine if the following sums converge or diverge. If they converge, then nd the value. (i) X1 i=0 1 2 n This is geometric with a= 1 and r= 1 2, A special case of the divergence theorem follows by specializing to the plane. Letting be a region in the plane with boundary , equation ( 1) then collapses to. (2) …, A divergent question is asked without an attempt to reach a direct or specific conclusion. It is employed to stimulate divergent thinking that considers a variety of outcomes to a certain proposal., Some examples of the 4-gradient as used in the d'Alembertian follow: ... More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a …, Most of the vector identities (in fact all of them except Theorem 4.1.3.e, Theorem 4.1.5.d and Theorem 4.1.7) are really easy to guess. Just combine the conventional linearity and product rules with the facts that, View Answer. Use the Divergence Theorem to calculate the surface integral \iint F. ds; that is calculate the flux of F across S: F (x, y, z) = xi - x^2j + 4xyzk, S is the surface of the solid bounded by the cyl... View Answer. Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux., The following examples illustrate the practical use of the divergence theorem in calculating surface integrals. Example 3 Let’s see how the result that was derived in Example 1 can be obtained by using the divergence theorem., As with Green's Theorem, and Stokes Theorem, there are ways to apply the divergence theorem indirectly. We illustrate with some examples. Example 1.4. Let S be the open cone z = p (x2 +y2) with z 6 3. Calculate Z Z S F~ ·dS~ for each of the following: (i) F~ = x~i +y~j +z~k (ii) F~ = x~i +y~j We consider each problem individually., The dot product, as best as I can guess, is meant to be a left tensor contraction so that $$ u\cdot(v\otimes w) = (u\cdot v)w. $$ Because the tensor product is ...