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Volume Differential

Volume differential, abbreviated as \(dV\), is an infinitesimally small element of volume used in physics, and engineering to analyze how quantities vary within a three-dimensional space.  It represents the smallest meaningful “piece” of volume over which a function can be evaluated before being summed or integrated to obtain totals such as mass, charge, fluid flow, or energy. 

Volume Differential (Cartesian Coordinates) Formula
\(  dV \;=\;  dx \cdot dy \cdot dz  \) 
Symbol English Metric
\(  dV \) = Volume Differential \( in^3 \)  \( mm^3 \) 
\( dx \) = Imperceptible Change in x-plane \( in \) \( mm \)
\( dy \) = Imperceptible Change in y-plane \( in \) \( mm \)
\( dz \) = Imperceptible Change in z-plane \( in \) \( mm \)

The volume differential is written as \(dV\), and its specific form depends on the coordinate system being used.  In Cartesian coordinates it takes the shape of a tiny rectangular box, while in cylindrical or spherical coordinates it becomes a small wedge-shaped or shell-shaped element.  By using the volume differential, integrals can accurately represent continuous physical quantities distributed throughout a region in space.

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Volume Differential (Cylindrical Coordinates) Formula
\( dV \;=\;  r \cdot dz \cdot dr \cdot d\theta  \) 
Symbol English Metric
\(  dV \) = Volume Differential \( in^3 \)  \( mm^2 \) 
\( r \) = Radius \( in \) \( mm \)
\( dz \) = Imperceptible Change in z-plane \( in \) \( mm \)
\( dr \) = Imperceptible Change in Radius  \(in\) \(mm\) 
\( d\theta \) = Imperceptible Change in Degree \(deg\) \(rad\)

 

 

 

 

 

 

 

 

 

Volume Differential (Spherical Coordinates) Formula
\(  dV \;=\;  r^2 \cdot  sin(\phi) \cdot dr \cdot d\phi \cdot d\theta  \) 
Symbol English Metric
\(  dV \) = Volume Differential \( in^3 \)  \( mm^2 \) 
\( r \) = Radial Distance \( in \) \( mm \)
\( sin(\phi) \) = sin Polar Angle \(deg\) \(rad\)
\( dr \) = Imperceptible Change in Radius Distance  \(in\) \(mm\) 
\( d\phi \) = Imperceptible Change in Polar Angle \(deg\) \(rad\)
\( d\theta \) = Imperceptible Change in Azimuthal Angle \(deg\)  \(rad\)