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# What is Surface integral: Definition and 260 Discussions
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.
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## Surface Integral of Vector Fields
Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = xy i + yz j + zx k S is the part of the paraboloid z = 8 − x^2 − y^2 that…
— solidet
— Thread
— Fields Integral Surface Surface integral Vector Vector fields
— Replies: 5
— Forum: Calculus and Beyond Homework Help
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## Surface Integral Homework: Compute F =
Homework Statement Compute ∫∫SF⋅dS where F =
— pyroknife
— Thread
— Integral Surface Surface integral
— Replies: 4
— Forum: Calculus and Beyond Homework Help
— P
## Surface Integral Problem: Solving for the Normal Vector and Double Integral
Homework Statement ∫∫s F⋅dS where F =
— pyroknife
— Thread
— Integral Surface Surface integral
— Replies: 2
— Forum: Calculus and Beyond Homework Help
## Surface integral in spherical coordinates question
Homework Statement Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 3c^2 within the paraboloid 2cz = x^2 + y^2 using spherical coordinates. (c is a constant) Homework Equations The Attempt at a Solution I converted all the x’s to
ho sin\phi cos\theta, y’s to
ho…
— ElijahRockers
— Thread
— Coordinates Integral Spherical Spherical coordinates Surface Surface integral
— Replies: 2
— Forum: Calculus and Beyond Homework Help
— K
## Surface integral over a sphere
Homework Statement Let \textbfr} be the position vector from the origin to a point (x,y,z) and r = |. Let \textbf{A} be a constant vector. Determine the value of the surface integral \int \textbf{A} \cdot \textbf{r} da over the surface of a sphere of radius R centered at the origin…
— Krayfish
— Thread
— Integral Sphere Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— B
## Surface Integral of a helicoid.
Homework Statement Evaluate \int\int_{S}\sqrt{1+x^{2}+y^{2}}dS where S is the helicoid: r(u,v)=ucos(v)i+usin(v)j+vk, with 0\leq u\leq1, 0\leq v\leq2\pi The Attempt at a Solution Here’s my attempt. I must have made some mistake, but I can’t figure out what. Any help would be…
— bjnartowt
— Thread
— Integral Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— P
## Surface Integral of a helicoid.
Homework Statement Evaluate \int\int_{S}\sqrt{1+x^{2}+y^{2}}dS where S is the helicoid: r(u,v)=ucos(v)i+usin(v)j+vk, with 0\leq u\leq1, 0\leq v\leq2\pi The Attempt at a Solution Here’s my attempt. I must have made some mistake, but I can’t figure out what. Any help would be…
— physicsnoob93
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— B
## Evaluating Surface Integral for f(x,y,z)=x^2+y^2+z^2
Homework Statement Evaluate the surface integral \int\int_S f(x,y,z)dS for f(x,y,z) = x^2 + y^2 + z^2, where S is the part of the cylinder x^2 + z^2 = 1 that lies between the planes y = -1 and y = 1, and for which z \geq 0. The Attempt at a Solution I parametrize the surface: let y = u, z =…
— bjnartowt
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— T
## Surface integral with vector integrand
If we integrate a vector field over a surface, \int_S \vec{F} \cdot \vec{dS}, we get the flux through that surface. What does it mean if the integrad were a vector instead of a vector dot product with the area element? It wouldn’t have the same physical meaning of flux anymore, so what would…
— tshafer
— Thread
— Integral Surface Surface integral Vector
— Replies: 4
— Forum: Calculus
— B
## Surface integral of a parabolic cylinder
Homework Statement Find the surface integral of the region defined by the parabolic cylinder y = x^2 where x > 0, and 0 < z < 4, and 0 < y < 3. The surface has positive orientation and the vector field is: F =
— bjnartowt
— Thread
— Cylinder Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— B
## Surface Integral of a Cylinder
Homework Statement Find the surface integral of the cylinder x^2 + z^2 = 9 lying above the rectangle with vertices (0,0,0), (4,0,0), (0,4,3), and (4,4,3). The surface has positive orientation and the vector field is: F =
— bjnartowt
— Thread
— Cylinder Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— S
## Surface integral with differential forms
Hi, I’m trying to solve a problem in David Bachman’s Geometric Approach to Differential Forms (teaching myself.) The problem is to integrate \omega = x dy \wedge dz + y dz \wedge dx + z dx \wedge dy over the unit hemisphere centered at the origin, using the parametric representation…
— samh
— Thread
— Differential Differential forms Forms Integral Surface Surface integral
— Replies: 1
— Forum: Differential Geometry
— H
## Surface Integral of a Cylinder
Let’s say I have a cylinder of height h and radius r that’s aligned with the z-axis. I need to find the surface integral of some function. So under normal circumstances I’d parameterize the surface in terms my two variables, which would be theta and z. Then I’d have to find the normal vector…
— hwill205
— Thread
— Cylinder Integral Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— S
## Surface integral or Divergence Theorem confused?
Homework Statement Find the Volume ∫∫ xy DA where R is the region bounded by by the line y=x-1 and the parabola y^2=2x+6. Homework Equations ∫∫ xy dx dy The Attempt at a Solution first i found the intersection of the above equations . which is (5,4) to (-1,-2) . then i simple…
— Saeed.z
— Thread
— Confused Divergence Divergence theorem Integral Surface Surface integral Theorem
— Replies: 2
— Forum: Calculus and Beyond Homework Help
— C
## Surface Integral over Half Shell
Homework Statement The problem is given in the attached picture. Basically, it is asking to determine the value of the surface integral for the given function, in region S, where S is the upper half of a spherical shell. Homework Equations The surface integral is defined as…
— corwest
— Thread
— Integral Shell Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— B
## Surface Integral Over Tetrahedron
I am trying to find the surface integral over the tetrahedron cut from the first octant by the plane 6x+3y+2z=6. Function is x^2. I am confused on the limits of integration. I am having troubles with this topic in general. I think I need to find the projection on the x-y plane; then I have z =…
— Bashyboy
— Thread
— Integral Surface Surface integral Tetrahedron
— Replies: 6
— Forum: Calculus and Beyond Homework Help
— S
## Surface Integral using Divergence Theorem
Homework Statement Evaluate the surface integral I = \int\int_S \vec{F}\cdot\vec{dS} using the divergence theorem, in the following cases. (i) \vec{F} = (xy, −y^2, z), and S is a closed surface of the region bounded by the planes y = 4 and z = 4 − x and the coordinate (x, y, z) planes. (ii)…
— Spoony
— Thread
— Divergence Divergence theorem Integral Surface Surface integral Theorem
— Replies: 1
— Forum: Calculus and Beyond Homework Help
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## Surface Integral of vector field
Homework Statement Let F =
— tony873004
— Thread
— Field Integral Surface Surface integral Vector Vector field
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— J
## Surface integral problem
Homework Statement Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i + y j + z^4 k S is the part of the cone z =…
— Jgoshorn1
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— B
## Surface Integral and Flux
Homework Statement Let S be the boundary of the solid region enclosed by the paraboloid z = 4 − x^2 − y^2 and the plane z = 2. Calculate the flux of the vector field F = (e^y + z)i + (x arctan(z^3))j + (3z + xy)k through S with outward orientation. Homework Equations The Attempt…
— bmxicle
— Thread
— Flux Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— E
## Surface Integral over a Cylinder.
Homework Statement Evaluate the surface integral, the double integral of F*ds, where F(x,y,z)=
— Emethyst
— Thread
— Cylinder Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— J
## Surface Integral of vector field
Hey there! I have a problem, namely: Evaluate the surface integral ∫∫ F.dS where F=
— Jimbo57
— Thread
— Field Integral Surface Surface integral Vector Vector field
— Replies: 9
— Forum: Calculus and Beyond Homework Help
— R
## Surface Integral of F on Cone S 0≤z≤4: Verify Divergence Theorem
Homework Statement Let F(x,y,z)=<2x,-3y,z> and let S be the cone z=sqrt(x^2+y^2) where 0\leq z\leq4 Verify the divergence theorem. Homework Equations Divergence Theorem: \int\int\int div(F)dV=\int\int F\cdot dS The Attempt at a Solution I have successfully solved the left hand…
— RBNeate
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— S
## Evaluating a Surface Integral: x^2 + y^2 + z^2 = a^2
Homework Statement Evaluate the surface integral ∫∫ z^2 dS where S is the surface given by x^2 + y^2 + z^2 = a^2. Homework Equations The Attempt at a Solution I think the easiest way to do this is via spherical coordinates, so I have: x = a sin φ cos θ y = a sin φ sin θ z = a cos…
— shaon0
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— S
## Surface integral to Lateral integral?
Homework Statement I have some working out my lecturer gave me and I’m trying to understand it. It’s about using Gauss’ Law to find the capacitance of a cylindrical capacitor of length L. It’s all good until the surface integral part, which is: \oint _S \vec{E} \cdot d\vec{S} =…
— Shaybay92
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— E
## Surface integral with vector integrand
Let S be the part of the ellipsoid x^2 + y^2 + 2z^2 = 1 with z >= 0. Evaluate \int\int _{\small{S}}(x^3i + y^3j + z^3k)\cdot dA. I know that the divergence theorem gives \int\int _{\small{S}}F\cdot dA = \int\int\int _{\small{V}}div F dV but I am not sure how to evaluate the left side directly…
— e12514
— Thread
— Integral Surface Surface integral Vector
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— B
## Surface Integral Homework: Compute F =
Homework Statement Compute ∫∫SF⋅dS where F =
— B18
— Thread
— Integral Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— K
## Surface integral with differential forms
Hi, I’m trying to solve a problem in David Bachman’s Geometric Approach to Differential Forms (teaching myself.) The problem is to integrate the scalar function f(x,y,z) = z^2 over the top half of the unit sphere centered at the origin, using the parametric representation phi: D -> S, where D…
— krcmd1
— Thread
— Differential Differential forms Forms Integral Surface Surface integral
— Replies: 8
— Forum: Differential Geometry
— S
## Surface Integral over a hemisphere
Homework Statement Find the surface integral of the hemisphere z = \sqrt{9 — x^2 — y^2} above the plane z = 2, if the density function is
ho (x,y,z) = z^2. Homework Equations Surface integral of a scalar field: \iint_T f(x,y,z) dS = \iint_R f(x,y,g(x,y)) \sqrt{z_x^2 + z_y^2 + 1} dA The…
— SpicyRamen
— Thread
— Hemisphere Integral Surface Surface integral
— Replies: 2
— Forum: Calculus and Beyond Homework Help
— C
## Surface Integral of a Cylinder
Homework Statement Let S denote the closed cylinder with bottom given by z=0, top given by z=4, and lateral surface given by the equation x^2 + y^2 = 9. Orient S with outward normals. Determine the indicated surface integral. The problem is ∫S x i \bullet n dS. (I wish I could post it in…
— chill_factor
— Thread
— Cylinder Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— C
## Surface Integral of a Cylinder
Homework Statement Let S denote the closed cylinder with bottom given by z=0, top given by z=4, and lateral surface given by the equation x^2 + y^2 = 9. Orient S with outward normals. Determine the integral: \int\int_{S} \vec{F} \cdot d\vec{S} where \vec{F} =
— chill_factor
— Thread
— Cylinder Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— C
## Surface Integral with Divergence Thm
Homework Statement Let S be the surface of the cylindrical solid bounded by x^2 + y^2 = 4, z = 0 , and z = 5. Evaluate the surface integral ∫∫ (y^2 + z) dS S where S is the surface of the solid. Homework Equations Divergence Thm: ∫∫ F\bulletn dS = ∫∫∫ divF dV The Attempt at a…
— chill_factor
— Thread
— Divergence Integral Surface Surface integral
— Replies: 8
— Forum: Calculus and Beyond Homework Help
— J
## Surface Integral (Divergence Theorem?)
Homework Statement See Attachment Homework Equations The Attempt at a Solution See Attachment. I thought the easiest way to do this would be to use the divergence theorem. However, I’m confused by the fact that there are two surfaces that the function is integrated over. How would…
— jumbogala
— Thread
— Divergence theorem Integral Surface Surface integral Theorem
— Replies: 2
— Forum: Calculus and Beyond Homework Help
— Z
## Surface Integral Assistance
Homework Statement I need to calculate a surface integral over S \int \int (x^2z + y^2z)dS where S is the hemisphere x^2 + y^2 + z^2 = 4, z \geq 0 Homework Equations I think I should be using this formula: \int \int_S f(x,y,z)dS = \int \int_D f(x,y, g(x,y)) \sqrt{(\frac{\partial…
— zooxanthellae
— Thread
— Assistance Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— P
## Surface Integral: Evaluating Double Integral of f.n dS
Homework Statement Evaluate the double integral of f.n dS where f = i + j + k and S is the surface z = x + y^2, 0 < x < 1, 0 < y < 2. Homework Equations The double integral of f.n dS The Attempt at a Solution First, i need to calculate the unit normal vector, n. I did this by...
— Precursor
— Thread
— Integral Surface Surface integral
— Replies: 7
— Forum: Calculus and Beyond Homework Help
— P
## Surface Integral: Evaluating f.n dS on 2x+y=6 Plane in 1st Octant
Homework Statement Evaluate the double integral of f.n dS where f = xi + yj and S is the plane 2x + y = 6 in the first octant. Homework Equations The double integral of f.n dS The Attempt at a Solution I found the normal vector to the plane, which is 2i + j. So f.n = 2x + y. Since…
— Precursor
— Thread
— Integral Surface Surface integral
— Replies: 4
— Forum: Calculus and Beyond Homework Help
— P
## Surface Integral: Evaluating f.n dS for f=yzi+zxj+xyk
Homework Statement Evaluate the double integral of f.n dS where f = yzi + zxj + xyk and S is the surface defined by the cylinder x^2 + y^2 = 1, bounded by the planes z = -1 and z = 1, and located in the first octant. Homework Equations The double integral of f.n dS The Attempt at a…
— Precursor
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— H
## Evaluating Surface Integral for S:y=1+x^2, 0≤x,z≤5
Homework Statement Evaluate the surface integral ∫∫S y dS where S is the surface y=1+x^2, 0 \leq x \leq 5, 0 \leq z \leq 5 Homework Equations r(x,z) = xi + (1+x^2)j + zk dS = |r_x \times r_z|dxdz = \sqrt{(2x)^2 + (-1)^2 + 0^2}dxdz The Attempt at a Solution After computing dS, I have…
— hadroneater
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— S
## Surface Integral — or Line Integral?
Homework Statement Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,0). Homework…
— Seda
— Thread
— Integral Line Line integral Surface Surface integral
— Replies: 4
— Forum: Calculus and Beyond Homework Help
— H
## Surface Integral, flux. Boundary and orientation
In solving the flux integral, I am having trouble finding out what I need to set as my bounds. I know that in order to find flux I need to take the integral of F dot N (dS) but I am not too clear on how to find the bounds or the orientation. For example in the following problem: F =
— h12x
— Thread
— Boundary Flux Integral Orientation Surface Surface integral
— Replies: 6
— Forum: Calculus and Beyond Homework Help
— S
## Surface Integral: Understanding Integration Limits in 3D Space
Homework Statement I’m reading through Schaum’s Outlines: Advanced Calculus to prep for taking PDE’s next semester, and I’m stuck on a surface integral problem (Edit: I think its a double integral problem, but the section is titled Surface Integrals so I’m not sure)…
— scottie_000
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— J
## Surface Integral: Integrating G(x, y, z) over a Parabolic Cylinder
Homework Statement Integrate G(x, y, z) = xyz over the parabolic cylinder y = x2, 0 \leq x \leq \frac{1}{2}, 0 \leq z \leq 3. Homework Equations The Attempt at a Solution I set up the integral in terms of x and z, but I’m not sure about the bounds, or if I set it up correctly…
— jens.w
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— L
## Surface integral with vector integrand
Let’s say I want to calculate ∫∫S F dS, where S is some surface and F is a vector field. My question is: when F = nx, where n is the unit normal vector to the surface and x is a position vector, can I write ∫∫S F dS = ∫∫S (nx) dS = x ∫∫S n dS ? That is, can I take the position vector x outside…
— Lajka
— Thread
— Integral Surface Surface integral Vector
— Replies: 4
— Forum: Calculus
— D
## Surface integral with differential forms
Hi, I have this problem about a surface integral. It says: Let \alpha and \beta be 1-forms and v and w vector fields. Show that: \iint_{\Sigma}<\alpha \wedge \beta, v \wedge w> = \int_{\Sigma} <\alpha,v><\beta,w>—<\alpha,w><\beta,v> where
— Damidami
— Thread
— Differential Differential forms Forms Integral Surface Surface integral
— Replies: 1
— Forum: Differential Geometry
— K
## Surface Integral and Flux
Homework Statement Let S be the boundary of the region enclosed by the cylinder y2 + z2 = 36 and the planes x = 0 and x + y = 9. Let F = [x, y + z, z − x]. Find the flux of F across S. The orientation of S is outward. Homework Equations Divergence Theorem: \int\int\int div(F) dV =…
— Kizaru
— Thread
— Flux Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— C
## Surface Integral of a helicoid.
Homework Statement Evaluate \int\int_{S}\sqrt{1+x^{2}+y^{2}}dS where S is the helicoid: r(u,v)=ucos(v)i+usin(v)j+vk, with 0\leq u\leq1, 0\leq v\leq2\pi Homework Equations The Attempt at a Solution Here’s my attempt. I must have made some mistake, but I can’t figure out what. Any…
— CuriousStudent
— Thread
— Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— J
## Surface Integral of a Vector Field
Homework Statement Given the vector field: v(x,y,z)=[4y,-x,2z] And the surface, S, which is the part of the paraboloid z=x^{2}+y^{2} that lies below the plane z=1, evaluate the integral: \int\int_{S} v \bullet dS 2. The attempt at a solution So, I’ve tried to solve this by using…
— Jizer
— Thread
— Field Integral Surface Surface integral Vector Vector field
— Replies: 5
— Forum: Calculus and Beyond Homework Help
— 1
## Surface Integral of a Circle
Homework Statement Given F = (-y/(x2+y2), x/(x2+y2), z2), calculate the surface integral of the curl of F (∇ x F) over the portion of the paraboloid z = x2 + y2 — 1 where z ≤ 0. Homework Equations The Attempt at a Solution I know that ∇ x F = (0, 0, 2/(x2+y2)). I also figured out…
— 1s1
— Thread
— Circle Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— B
## Surface Integral Homework: ∫∫σ3×2 + 3y2 + 3z2dS
Homework Statement Evaluate ∫∫σ3×2 + 3y2 + 3z2dS where σ is the part of the cylinder x2 + y2 = 4 between the planes z = 0 and z = 1 together with the top and bottom circular caps. Homework Equations The Attempt at a Solution So the surface is a cylinder with height 1 and radius…
— BeBattey
— Thread
— Integral Surface Surface integral
— Replies: 2
— Forum: Calculus and Beyond Homework Help
## Surface Integral: Understanding the Concept
Hi, I’m trying to grasp the idea behind Surface integrals. As I understand it, right now my objective is to find the «total» of the vector field F(x,y,z) over the surface S. To do this, I need to take the dot product of F and the normal vector n, and integrate this over the surface S. The…
— jinksys
— Thread
— Integral Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— T
## Surface Integral Help: Calculating \int\int_S z dS
Homework Statement Evaluate the surface integral. \int\int_S z dS S is the surface x = y + 2z^2, 0 \leq y \leq 1, 0 \leq z \leq 3. Homework Equations \int\int_S f(x, y, z)dS = \int\int_D f(r(u, v))|r_u \times r_v|dA The Attempt at a Solution So, I’m having a tough time…
— tangibleLime
— Thread
— Integral Surface Surface integral
— Replies: 3
— Forum: Calculus and Beyond Homework Help
— J
## Surface Integral of a Sphere
Homework Statement Evaluate the surface integral of the upper half of the sphere with radius=a. 2. Relevant information I know that I need to use the equation \int\int\int x^2 dxdydz but I’m not really sure how to proceed with this problem 3. Attempt at a solution I know that x^2…
— jhosamelly
— Thread
— Integral Sphere Surface Surface integral
— Replies: 7
— Forum: Calculus and Beyond Homework Help
— T
## Surface Integral over a Hemisphere
Homework Statement Surface integral of the hemisphere z=sqrt(a^2-x^2-y^2) Homework Equations da = dxdy / |n . k| The Attempt at a Solution the normal to the surface is ( ∂z/∂x , ∂z/∂y , -1 ) = ( x/sqrt(a^2-x^2-y^2) , y/sqrt(a^2-x^2-y^2) , -1 ) , the dot product of this with k is -1, so…
— theneedtoknow
— Thread
— Hemisphere Integral Surface Surface integral
— Replies: 1
— Forum: Calculus and Beyond Homework Help
— T
## Surface Integral with Divergence Thm
Homework Statement Let W be the solid cylindrical region defined by x^2 + y^2 ≤ 16, 0 ≤ z ≤ π/2, and let V = (x-tan yz)i + (y- e^xz)j + (3z + ln xy)k. Find the flux of V outward the region through the boundary of W. Homework Equations Divergence Thm: \int\int\int div V dV = \int\int V dS…
— theneedtoknow
— Thread
— Divergence Integral Surface Surface integral
— Replies: 5
— Forum: Calculus and Beyond Homework Help
— L
## Surface integral with vector integrand
Let’s say I want to calculate \iint \mathbf{F} \cdot \mathbf{dS}, where S is some surface and \mathbf{F} is a vector field. My question is: when \mathbf{F} = n \mathbf{x}, where n is the unit normal vector to the surface and \mathbf{x} is a position vector, can I write \iint \mathbf{F} \cdot…
— Lajka
— Thread
— Integral Surface Surface integral Vector
— Replies: 1
— Forum: Calculus
Дизайн маленького кабинета в квартире: идеи и решения
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Отдавайте предпочтение мебели с небольшими габаритами и многофункциональным дизайном. Например, письменный стол с выдвижными ящиками или откидной столешницей поможет сэкономить место. Для сидения рассмотрите узкие стулья без подлокотников или табуреты, которые можно убрать под стол.
Оптимизируйте освещение
Используйте точечные светильники или настенные бра, чтобы не занимать пространство напольными лампами. Если естественного света недостаточно, добавьте настольную лампу с регулируемым кронштейном, чтобы направлять свет точно в нужную зону.
Создайте акцент на одной стене. Покрасьте её в яркий цвет или используйте обои с мелким узором, чтобы добавить глубину интерьеру. Это отвлечёт внимание от небольших размеров комнаты и сделает её визуально просторнее.
Храните документы и мелочи в коробках или органайзерах, которые можно разместить на полках или в нишах. Это поможет избежать беспорядка и сохранить рабочую зону аккуратной. Используйте встроенные шкафы или модульные системы, чтобы максимально задействовать каждый уголок.
Добавьте зеркала или стеклянные элементы. Они отражают свет и создают иллюзию большего пространства. Например, зеркало над столом или стеклянная столешница сделают кабинет светлее и воздушнее.
Минимизируйте декор. Выберите один-два акцента, таких как стильные часы или небольшое растение, чтобы не перегружать интерьер. Это сохранит ощущение простора и уюта.
