Name: flux through closed surfaces
Uploaded: 2026-06-08T09:39:08.515Z
Duration: 1 min 20 s
Channel: JoVE

In vector calculus, the total flow of a vector field through a surface is called flux. For closed three-dimensional shapes, calculating this at every boundary point is often a geometric challenge.Given a vector field and a closed surface, the divergence theorem equates the total flux of the vector field through the closed surface to the volume integral of the field&#x2019;s divergence within the enclosed region.This theorem shows its utility when applied to the electric field of a central point charge extending through the hollow region bounded between an inner sphere and an irregular outer boundary.The total surface integral of this hollow region is equal to the difference between the outer boundary&#x27;s flux and the inner boundary&#x27;s flux due to their opposing normal vectors.Because no charge exists within this hollow region, the divergence of the electric field within this specific volume is exactly zero. So the outward flux through the irregular outer boundary is exactly equal to the flux through the inner sphere.This proves that net flux depends entirely on the enclosed charge, completely independent of the outer surface&#x27;s shape.

divergence theorem in 3d space

In vector calculus, flux measures the total flow of a vector field through a surface. For a closed surface in three-dimensional space, this means measuring how much of the field passes outward through every point on the boundary. Directly calculating this flux can be difficult when the surface has a complicated or irregular shape. The Divergence Theorem provides a powerful alternative by relating surface flux to behavior inside the enclosed region.The Divergence Theorem states that the outward flux of a vector field through a closed surface is equal to the triple integral of the divergence of the field over the solid region enclosed by that surface. In symbolic form,\begin{equation*}\iiint_V (\nabla \cdot \mathbf{F})\, dV=\iint_S \mathbf{F} \cdot \mathbf{n}\, dS\end{equation*}Here, F is the vector field, S is the closed surface, n is the outward unit normal vector, and E is the enclosed solid region. The divergence measures the extent to which the vector field behaves like a source or sink at each point.The theorem is especially useful for electric fields produced by point charges. Consider the hollow region between a small inner sphere surrounding a central charge and a complex outer boundary. Within this hollow region, there are no charges. Therefore, the divergence of the electric field is zero everywhere in that region.Applying the Divergence Theorem shows that the total outward flux through the boundary of the hollow region must also be zero. This means the flux through the irregular outer surface is exactly balanced by the flux through the inner sphere, with orientation taken into account. As a result, the total electric flux through the outer boundary is independent of its shape. Whether the surface is spherical or irregular, the flux depends only on the net charge enclosed, not on the geometry of the boundary.

Divergence Theorem in 3D Space

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Vector Calculus and Theorems

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Théorème de divergence dans l’espace 3D

flux through closed surfaces

Flux through a closed surface in vector calculus measures how much of a vector field passes outward across the boundary. When the surface in three-dimensional space has an irregular shape, calculating that outward flow directly can be difficult. The Divergence Theorem gives a faster way to connect the surface flux to what is happening inside the enclosed region.

The theorem states that the outward flux of a vector field through a closed surface equals the triple integral of the divergence over the solid region inside it. In symbolic form, \begin{equation*} \iiint_V (\nabla \cdot \mathbf{F})\, dV =\iint_S \mathbf{F} \cdot \mathbf{n}\, dS \end{equation*} Here, F is the vector field, S is the closed surface, n is the outward unit normal vector, and E is the enclosed solid region. The divergence shows how much a field acts like a source or sink at a point.

The theorem is especially useful for electric fields from point charges. Imagine a hollow region between a small inner sphere around a central charge and a complex outer boundary. There are no charges in that hollow region, so the divergence of the electric field is zero everywhere inside it. Using the Divergence Theorem, the total outward flux through the boundary of the hollow region must also be zero.

That means the flux through the irregular outer surface is balanced by the flux through the inner sphere, once orientation is taken into account. As a result, the total electric flux through the outer boundary does not depend on the shape of the surface. Whether the boundary is spherical or irregular, the flux depends only on the net charge enclosed, not on the geometry of the boundary.

Flux Through Closed Surfaces

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Flux through a closed surface in vector calculus measures how much of a vector field passes outward across the boundary. When the surface in three-dimensional space has an irregular shape, calculating that outward flow directly can be difficult. The Divergence Theorem gives a faster way to connect the surface flux to what is happening inside the enclosed region. The theorem states that the outward flux of a vector field through a closed surface equals the triple integral of the divergence over...

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key terms and concepts

Learn how the Divergence Theorem links outward flux through a closed surface to the divergence inside, with an electric field example using point charges.

Flux Through Closed Surfaces | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Flux Through Closed Surfaces

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