The Central Idea
Flux measures the component of a vector field that passes through a curve or surface.
The key geometric idea is the dot product
\[ \boxed{ \mathbf{F}\cdot\mathbf{n} }. \]Here \(\mathbf{F}\) is the field vector and \(\mathbf{n}\) is a chosen normal direction to the boundary.
The dot product keeps only the part of the field pointing through the boundary.
Along versus Through
Vector calculus uses two closely related but different kinds of line or surface measurements.
A tangential integral measures how much the field points along a path.
A flux integral measures how much the field points through a boundary.
The difference is determined by whether we compare the field with a tangent direction or a normal direction.
Normal Vectors
A normal vector is perpendicular to a curve or surface at a point.
Flux uses this perpendicular direction because crossing a boundary is measured relative to the direction normal to it.
If \(\mathbf{n}\) is a unit normal, then
\[ \mathbf{F}\cdot\mathbf{n} \]is the signed normal component of the field.
The Dot Product Controls Flux
Recall
\[ \mathbf{F}\cdot\mathbf{n} = \|\mathbf{F}\| \|\mathbf{n}\| \cos\theta. \]If \(\mathbf{n}\) is a unit normal,
\[ \mathbf{F}\cdot\mathbf{n} = \|\mathbf{F}\|\cos\theta. \]Therefore, flux depends on how strongly the field aligns with the normal direction.
Maximum Positive Crossing
If the field points directly in the chosen normal direction, then
\[ \theta=0. \]Hence,
\[ \mathbf{F}\cdot\mathbf{n} = \|\mathbf{F}\|. \]This gives the largest positive normal contribution for that field magnitude.
Zero Crossing Contribution
If the field is tangent to the boundary, then
\[ \theta=90^\circ. \]Therefore,
\[ \mathbf{F}\cdot\mathbf{n}=0. \]The field may be strong, but none of it crosses the boundary locally.
Negative Flux Contribution
If the field points opposite the chosen normal direction, then
\[ \mathbf{F}\cdot\mathbf{n}<0. \]Negative flux does not mean “negative amount of field.”
It means the crossing direction is opposite the selected orientation.
Orientation Is Essential
A curve or surface generally has two possible normal directions.
Choosing one normal rather than the other reverses the sign of flux:
\[ \mathbf{F}\cdot(-\mathbf{n}) = -\mathbf{F}\cdot\mathbf{n}. \]Therefore, flux is an oriented quantity.
Closed Boundaries Have Natural Orientations
For a closed curve in the plane, the standard flux orientation is outward.
For a closed surface in three dimensions, the standard orientation is also outward.
This makes positive total flux correspond naturally to net outward passage.
Flux through a Small Flat Surface
Suppose a constant vector field \(\mathbf{F}\) crosses a flat surface of area \(A\) with constant unit normal \(\mathbf{n}\).
Then the flux is
\[ \boxed{ \Phi = \mathbf{F}\cdot\mathbf{n}\,A }. \]This is the basic model from which the integral definition grows.
Why an Integral Is Needed
Real fields may vary from point to point.
Real surfaces may curve, so their normal directions also change from point to point.
We therefore divide the boundary into tiny pieces, approximate the field and normal as locally constant, and add the contributions.
The limiting sum becomes a flux integral.
Surface Flux in Three Dimensions
If \(S\) is an oriented surface with unit normal \(\mathbf{n}\), then flux is
\[ \boxed{ \iint_S \mathbf{F}\cdot\mathbf{n}\,dS }. \]The expression
\[ \mathbf{F}\cdot\mathbf{n} \]gives the normal component, while \(dS\) represents a tiny area element.
Vector Area Element
The oriented area element is often written
\[ d\mathbf{S} = \mathbf{n}\,dS. \]Then the flux integral becomes
\[ \boxed{ \iint_S \mathbf{F}\cdot d\mathbf{S} }. \]This notation packages area and orientation together.
Flux through a Plane
Suppose
\[ \mathbf{F} = \langle 2,0,0\rangle \]is constant and a flat surface of area \(5\) has unit normal
\[ \mathbf{n}=\langle 1,0,0\rangle. \]Then
\[ \mathbf{F}\cdot\mathbf{n}=2. \]Therefore,
\[ \boxed{\Phi=2(5)=10}. \]Reverse the Orientation
If the same surface is oriented by
\[ \mathbf{n}=\langle -1,0,0\rangle, \]then
\[ \mathbf{F}\cdot\mathbf{n}=-2. \]Therefore,
\[ \boxed{\Phi=-10}. \]The physical crossing is unchanged; only the orientation convention has reversed.
Flux through a Surface Parallel to the Field
Suppose
\[ \mathbf{F}=\langle 1,0,0\rangle \]and the surface normal is
\[ \mathbf{n}=\langle 0,1,0\rangle. \]Then
\[ \mathbf{F}\cdot\mathbf{n}=0. \]The field runs parallel to the surface and contributes zero flux through it.
Flux through a Curve in Two Dimensions
In the plane, a curve can act as a one-dimensional boundary.
For a vector field
\[ \mathbf{F}(x,y), \]the flux through an oriented curve \(C\) is
\[ \boxed{ \int_C \mathbf{F}\cdot\mathbf{n}\,ds }. \]Here \(\mathbf{n}\) is a chosen unit normal to the curve.
Arc Length Element
The symbol
\[ ds \]represents an infinitesimal arc length along the curve.
Therefore,
\[ \mathbf{F}\cdot\mathbf{n}\,ds \]represents the local normal field contribution across a tiny curve segment.
Flux versus Circulation in the Plane
For a planar curve:
circulation uses the tangent direction,
\[ \int_C \mathbf{F}\cdot\mathbf{t}\,ds, \]while flux uses the normal direction,
\[ \int_C \mathbf{F}\cdot\mathbf{n}\,ds. \]The same vector field can therefore be analyzed from two perpendicular viewpoints.
Parameterizing a Plane Curve
Let
\[ \mathbf{r}(t) = \langle x(t),y(t)\rangle. \]Then
\[ \mathbf{r}'(t) = \langle x'(t),y'(t)\rangle \]is tangent to the curve.
A perpendicular vector is
\[ \langle y'(t),-x'(t)\rangle \]or its negative.
A Convenient Planar Flux Formula
If the chosen oriented normal-times-length element is
\[ \mathbf{n}\,ds = \langle y'(t),-x'(t)\rangle\,dt, \]then for
\[ \mathbf{F}=\langle P,Q\rangle, \]the flux becomes
\[ \boxed{ \int \left( P\,y' - Q\,x' \right)\,dt }. \]Reversing the chosen normal reverses the sign.
Flux through a Vertical Line Segment
Consider the constant field
\[ \mathbf{F} = \langle 3,1\rangle \]and the vertical segment from \((0,0)\) to \((0,4)\).
Choose the rightward unit normal
\[ \mathbf{n}=\langle 1,0\rangle. \]Then
\[ \mathbf{F}\cdot\mathbf{n}=3. \]Since the segment length is \(4\),
\[ \boxed{\Phi=12}. \]Physical Meaning in Fluid Flow
If
\[ \mathbf{v} \]is a fluid velocity field, then
\[ \mathbf{v}\cdot\mathbf{n} \]gives the normal component of fluid velocity through the surface.
Integrating over area gives a volumetric flow rate when the units and model are appropriate.
Volumetric Flow Rate
If velocity is measured in
\[ \text{m/s} \]and area in
\[ \text{m}^2, \]then
\[ \iint_S \mathbf{v}\cdot\mathbf{n}\,dS \]has units
\[ \boxed{\text{m}^3/\text{s}}. \]These are units of volume per time.
Mass Flux
If a fluid has density
\[ \rho \]and velocity
\[ \mathbf{v}, \]then the mass-flow vector field may be written
\[ \rho\mathbf{v}. \]The mass flux through a surface is
\[ \boxed{ \iint_S \rho\mathbf{v}\cdot\mathbf{n}\,dS }. \]Electric Flux
For an electric field
\[ \mathbf{E}, \]electric flux through a surface is
\[ \boxed{ \iint_S \mathbf{E}\cdot\mathbf{n}\,dS }. \]The flux measures how strongly the electric field passes through the oriented surface.
Heat Flux
If
\[ \mathbf{q} \]is a heat-flux density vector, then the net heat-transfer rate through a surface is represented by
\[ \boxed{ \iint_S \mathbf{q}\cdot\mathbf{n}\,dS }. \]The sign depends on the chosen orientation.
Closed-Surface Flux
For a closed surface \(S\), outward flux is commonly written
\[ \boxed{ \oiint_S \mathbf{F}\cdot\mathbf{n}\,dS }. \]The circle on the integral sign emphasizes that the surface is closed.
Outward orientation is usually assumed unless stated otherwise.
Positive Net Outward Flux
If
\[ \oiint_S \mathbf{F}\cdot\mathbf{n}\,dS>0, \]then more field passes outward than inward through the closed boundary.
This is consistent with net source-like behavior inside the enclosed region.
Negative Net Outward Flux
If
\[ \oiint_S \mathbf{F}\cdot\mathbf{n}\,dS<0, \]then the inward contribution exceeds the outward contribution.
This is consistent with net sink-like behavior inside the region.
Zero Net Flux
A closed surface can have
\[ \oiint_S \mathbf{F}\cdot\mathbf{n}\,dS=0 \]even when the field is nonzero everywhere on the surface.
Outward and inward contributions may cancel.
Flux and Divergence
Divergence measures local source or sink behavior.
Flux measures field passage through a boundary.
These are linked by the divergence theorem:
\[ \boxed{ \oiint_{\partial V} \mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV }. \]This theorem converts a global boundary measurement into an accumulated local volume measurement.
What the Divergence Theorem Says Conceptually
Add up the local divergence throughout a volume.
The result equals the total outward flux through the enclosing surface.
In words:
\[ \boxed{ \text{total internal source strength} = \text{net outward boundary flux} }. \]A Constant-Divergence Example
Consider
\[ \mathbf{F}(x,y,z) = \langle x,y,z\rangle. \]Then
\[ \nabla\cdot\mathbf{F}=3. \]If a volume \(V\) has volume
\[ \operatorname{Vol}(V), \]then the divergence theorem gives
\[ \boxed{ \text{outward flux} = 3\operatorname{Vol}(V) }. \]Flux through a Sphere in a Radial Field
Suppose
\[ \mathbf{F}(\mathbf{r}) = c\frac{\mathbf{r}}{\|\mathbf{r}\|^3} \]on a sphere of radius \(R\).
The outward unit normal is
\[ \mathbf{n} = \frac{\mathbf{r}}{R}. \]On the sphere,
\[ \|\mathbf{r}\|=R. \]Compute the Radial Normal Component
On the sphere,
\[ \mathbf{F} = c\frac{\mathbf{r}}{R^3}. \]Therefore,
\[ \mathbf{F}\cdot\mathbf{n} = c\frac{\mathbf{r}}{R^3} \cdot \frac{\mathbf{r}}{R} = c\frac{R^2}{R^4} = \frac{c}{R^2}. \]Complete the Sphere Flux
The surface area of a sphere is
\[ 4\pi R^2. \]Since the normal component is constant over the sphere,
\[ \Phi = \frac{c}{R^2} (4\pi R^2). \]Therefore,
\[ \boxed{ \Phi=4\pi c }. \]The flux is independent of the sphere radius.
Why Radius Independence Is Important
The inverse-square decrease in field strength exactly balances the \(R^2\) increase in spherical surface area.
Therefore, the same total radial flux passes through every centered sphere surrounding the source.
This geometry underlies major inverse-square field laws.
Flux Can Be Distributed Unevenly
Total flux does not require the field to cross every part of a surface equally.
One region may have strong positive crossing while another has weak or negative crossing.
The integral adds all of the local signed contributions.
Flux Is an Accumulated Quantity
The local quantity
\[ \mathbf{F}\cdot\mathbf{n} \]describes normal field intensity at one point.
The integral
\[ \iint_S \mathbf{F}\cdot\mathbf{n}\,dS \]accumulates this information across the whole surface.
Flux Density versus Total Flux
The expression
\[ \mathbf{F}\cdot\mathbf{n} \]can be interpreted as a local normal flux density when the physical model supports that interpretation.
Multiplying by area and integrating produces the total flux.
Open Surfaces Need an Explicit Orientation
A closed surface has a natural outward direction.
An open surface does not.
For an open surface, the problem must specify or imply a choice such as:
- upward;
- downward;
- leftward;
- rightward;
- toward a particular region.
Upward and Downward Orientation
For a graph
\[ z=g(x,y), \]one may choose an upward-pointing normal or a downward-pointing normal.
These two choices differ by a sign.
The corresponding flux integrals are negatives of one another.
Surface Parameterization
A surface can be parameterized by
\[ \mathbf{r}(u,v). \]Two tangent vectors are
\[ \mathbf{r}_u \]and
\[ \mathbf{r}_v. \]Their cross product provides an oriented normal-area vector:
\[ \boxed{ \mathbf{r}_u\times\mathbf{r}_v }. \]Parameterized Surface Flux Formula
If
\[ \mathbf{r}(u,v) \]parameterizes the surface, then one orientation gives
\[ \boxed{ \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot \left( \mathbf{r}_u\times\mathbf{r}_v \right) \,du\,dv }. \]Reversing the cross-product order reverses the orientation.
Why the Cross Product Works
The vector
\[ \mathbf{r}_u\times\mathbf{r}_v \]is perpendicular to the surface.
Its magnitude gives the local area-scaling factor.
Therefore, it packages both normal direction and surface area into one vector quantity.
Flux through a Graph Surface
For a surface
\[ z=g(x,y), \]an upward-oriented normal-area vector can be written
\[ \boxed{ \langle -g_x,-g_y,1\rangle\,dx\,dy }. \]Therefore, flux can be computed over the projected region in the \(xy\)-plane.
Downward Orientation for a Graph
The downward-oriented normal-area vector is
\[ \boxed{ \langle g_x,g_y,-1\rangle\,dx\,dy }. \]It is exactly the negative of the upward-oriented choice.
Flux through Piecewise Surfaces
A boundary may be composed of several surface pieces.
The total flux is the sum:
\[ \boxed{ \Phi_{\text{total}} = \Phi_1+\Phi_2+\cdots+\Phi_n }. \]Each piece must use an orientation consistent with the overall boundary orientation.
Symmetry Can Simplify Flux
If the field and surface have matching symmetry, the normal component may be constant or may cancel in predictable ways.
Examples include:
- radial fields on spheres;
- uniform fields through flat faces;
- odd contributions canceling across symmetric regions.
Flux through a Closed Box
A rectangular box has six faces.
Total outward flux is found by adding the outward flux through all six faces.
When the divergence is simple, the divergence theorem may replace six separate surface calculations with one volume integral.
Direct Flux versus Divergence Theorem
There are often two strategies for a closed surface:
- integrate \(\mathbf{F}\cdot\mathbf{n}\) directly over the boundary;
- integrate \(\nabla\cdot\mathbf{F}\) over the enclosed volume.
The better method depends on the geometry and algebra.
Flux through a Closed Curve in the Plane
In two dimensions, outward flux through a closed curve can also be linked to divergence over the enclosed region.
A planar form of Green's theorem gives
\[ \boxed{ \oint_C \mathbf{F}\cdot\mathbf{n}\,ds = \iint_R \nabla\cdot\mathbf{F}\,dA }. \]This is the two-dimensional analogue of the divergence theorem.
Local and Global Flux
The quantity
\[ \mathbf{F}\cdot\mathbf{n} \]is local.
The surface integral is global over the selected boundary.
Flux is therefore a natural bridge between local vector-field direction and global transport.
Flux Depends on the Boundary
Even for the same vector field, different curves or surfaces can have different flux values.
Flux is not a property of the field alone.
It depends on:
- the field;
- the chosen boundary;
- the orientation.
Flux through the Same Surface Can Change with Time
If
\[ \mathbf{F}(x,y,z,t) \]varies with time, then the flux through a fixed surface may also vary:
\[ \Phi(t) = \iint_S \mathbf{F}(x,y,z,t) \cdot\mathbf{n}\,dS. \]Time dependence can therefore change transport even when the boundary does not move.
Moving Boundaries Add More Complexity
In some applications, both the field and the boundary move with time.
Then transport depends on the changing field, changing geometry, and changing orientation.
Such problems lead to more advanced transport theorems.
Flux and Conservation Laws
Many conservation laws have the general form
\[ \boxed{ \text{change inside} = \text{production inside} - \text{net outward flux} }. \]This structure appears in models of mass, energy, charge, and other transported quantities.
Why Flux Is Central to Applied Mathematics
Flux answers practical questions such as:
- How much fluid passes through a pipe cross-section?
- How much heat crosses a wall?
- How much electric field passes through a closed surface?
- How much transported material leaves a region?
The same integral structure serves many physical interpretations.
Common Error: Using the Tangent Instead of the Normal
Flux measures crossing, so it uses a normal direction.
A tangent-direction integral measures along-path behavior instead.
Always ask whether the problem says along or through.
Common Error: Ignoring Orientation
Reversing the normal reverses the sign of the flux.
Therefore, an answer without a clear orientation may be incomplete or ambiguous.
Common Error: Assuming Positive Flux Means Large Field Magnitude
Flux depends on both magnitude and angle.
A large field tangent to a surface can contribute zero flux.
A smaller field aligned with the normal can contribute more.
Common Error: Assuming Zero Flux Means Zero Field
Zero flux can occur because:
- the field is tangent to the boundary;
- positive and negative contributions cancel;
- the field itself is zero;
- symmetry produces cancellation.
The integral value alone does not determine the pointwise field.
Common Error: Forgetting the Area Element
The quantity
\[ \mathbf{F}\cdot\mathbf{n} \]is a local normal component.
Total flux requires integration against
\[ dS \]or, in two dimensions,
\[ ds. \]Common Error: Using a Nonunit Normal without Compensation
If the formula is written as
\[ \mathbf{F}\cdot\mathbf{n}\,dS, \]then \(\mathbf{n}\) should be a unit normal.
If a nonunit normal is used, its magnitude must be incorporated correctly into the area element.
Parameterized formulas handle this automatically through the cross product.
Common Error: Applying the Divergence Theorem to an Open Surface
The divergence theorem relates flux through a closed surface to a volume integral.
An open surface must be closed first, or handled directly, depending on the problem.
Common Error: Ignoring Singularities inside a Closed Surface
The usual divergence-theorem hypotheses require the field to be sufficiently smooth on the volume being considered.
A singularity inside the region can invalidate a naive application.
Domain analysis comes before theorem application.
A Reliable Procedure for Simple Flux
- Identify the vector field.
- Identify the curve or surface.
- Determine the required orientation.
- Find the appropriate normal direction.
- Compute \(\mathbf{F}\cdot\mathbf{n}\).
- Multiply by length or area if the quantities are constant.
- Otherwise integrate over the boundary.
- Interpret the sign and units.
A Reliable Procedure for Surface Flux
- Choose a parameterization or surface representation.
- Construct an oriented normal-area element.
- Evaluate the field on the surface.
- Take the dot product with the normal-area vector.
- Set the correct parameter domain.
- Integrate.
- Check orientation and units.
A Reliable Procedure for Closed-Surface Flux
- Verify that the surface is closed.
- Use outward orientation unless told otherwise.
- Inspect the field for singularities inside the region.
- Decide whether direct surface integration or the divergence theorem is simpler.
- Compute carefully.
- Interpret positive flux as net outward passage and negative flux as net inward passage.
Example: Constant Field through a Rectangle
Let
\[ \mathbf{F} = \langle 0,0,6\rangle \]and let a horizontal rectangle have area
\[ A=8. \]Choose upward orientation:
\[ \mathbf{n} = \langle 0,0,1\rangle. \]Then
\[ \mathbf{F}\cdot\mathbf{n}=6. \]Therefore,
\[ \boxed{\Phi=48}. \]Example: Same Rectangle, Downward Orientation
With downward orientation,
\[ \mathbf{n} = \langle 0,0,-1\rangle. \]Then
\[ \mathbf{F}\cdot\mathbf{n}=-6. \]Therefore,
\[ \boxed{\Phi=-48}. \]Example: Outward Flux from a Sphere
Let
\[ \mathbf{F}(x,y,z) = \langle x,y,z\rangle \]and let \(S\) be a sphere of radius \(R\).
The divergence is
\[ \nabla\cdot\mathbf{F}=3. \]By the divergence theorem,
\[ \Phi = \iiint_V 3\,dV. \]Since
\[ \operatorname{Vol}(V) = \frac{4}{3}\pi R^3, \]we obtain
\[ \boxed{ \Phi=4\pi R^3 }. \]Practice 1: Local Normal Component
Let
\[ \mathbf{F}=\langle 3,4\rangle \]and
\[ \mathbf{n}=\langle 1,0\rangle. \]Find \(\mathbf{F}\cdot\mathbf{n}\).
Practice 2: Tangential Field
If a field is tangent to a surface at a point, what is its local flux contribution there?
Practice 3: Orientation Reversal
If the flux through a surface is \(12\) with one orientation, what is it with the opposite orientation?
Practice 4: Constant Field through a Flat Surface
A constant field has
\[ \mathbf{F}\cdot\mathbf{n}=5 \]across a flat surface of area \(7\).
Find the flux.
Practice 5: Zero Net Flux
Does zero net flux through a closed surface imply that the vector field is zero everywhere on the surface?
Practice 6: Fluid Units
If a velocity field is measured in meters per second and area is measured in square meters, what units does volumetric flux have?
Practice 7: Divergence Theorem
Let
\[ \mathbf{F}(x,y,z)=\langle x,y,z\rangle \]and let \(V\) be a region of volume \(10\).
Find the outward flux through its closed boundary.
Practice 8: Flux through a Vertical Segment
Let
\[ \mathbf{F}=\langle 2,-1\rangle \]and let a vertical line segment have length \(6\).
Using the rightward unit normal, find the flux.
Practice 9: Radial Sphere Flux
Suppose
\[ \mathbf{F} = \frac{\mathbf{r}}{\|\mathbf{r}\|^3}. \]Find the outward flux through any sphere centered at the origin.
Practice 10: Through or Along?
Which geometric direction is used for flux: the tangent or the normal?
Practice Answers
Practice 1
\[ \mathbf{F}\cdot\mathbf{n} = \langle 3,4\rangle \cdot \langle 1,0\rangle = \boxed{3}. \]Practice 2
A tangent field is perpendicular to the normal, so
\[ \boxed{ \mathbf{F}\cdot\mathbf{n}=0 }. \]Practice 3
Reversing orientation reverses the sign:
\[ \boxed{-12}. \]Practice 4
\[ \Phi = 5(7) = \boxed{35}. \]Practice 5
No. Positive and negative contributions may cancel, or the field may be tangent over portions of the surface.
Practice 6
\[ \frac{\text{m}}{\text{s}} \cdot \text{m}^2 = \boxed{ \frac{\text{m}^3}{\text{s}} }. \]Practice 7
The divergence is
\[ 1+1+1=3. \]Therefore,
\[ \Phi = 3(10) = \boxed{30}. \]Practice 8
With rightward normal
\[ \mathbf{n}=\langle 1,0\rangle, \]we have
\[ \mathbf{F}\cdot\mathbf{n}=2. \]Therefore,
\[ \Phi=2(6)=\boxed{12}. \]Practice 9
This is the case \(c=1\) in the radial inverse-square calculation, so
\[ \boxed{ \Phi=4\pi }. \]Practice 10
Flux uses the
\[ \boxed{\text{normal direction}}. \]What to Remember
Flux measures how much of a vector field passes through an oriented boundary.
The local geometric quantity is
\[ \boxed{ \mathbf{F}\cdot\mathbf{n} }. \]In two dimensions,
\[ \boxed{ \int_C \mathbf{F}\cdot\mathbf{n}\,ds } \]measures flux through a curve.
In three dimensions,
\[ \boxed{ \iint_S \mathbf{F}\cdot\mathbf{n}\,dS } \]measures flux through a surface.
- flux uses normal rather than tangent directions;
- orientation determines the sign;
- tangent fields contribute zero local flux;
- positive and negative contributions can cancel;
- closed-surface flux is naturally oriented outward;
- the divergence theorem connects total outward flux with accumulated divergence;
- flux appears in fluid flow, electricity, heat transfer, and conservation laws;
- domain restrictions and singularities must be checked before applying global theorems.
The central geometric habit is: to measure what crosses a boundary, project the field onto the boundary's normal direction and accumulate that contribution across the boundary.
Where We Go Next
Flux gives an accumulated measure of how a vector field crosses a boundary. The remaining work in this section turns from symbolic field formulas to practical representations.
The next lesson, Numerical and Visual Field Representations, develops how scalar and vector fields are sampled, approximated, plotted, and interpreted when exact formulas are unavailable or too complicated to use directly.