Question about vectors and dot product.

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SUMMARY

The discussion addresses two key mathematical concepts involving vector operations and calculus. It confirms that the expression -E dotted with (A + B) is indeed equal to -E.A - E.B, indicating that the minus sign applies to both terms. Additionally, it establishes that under certain conditions, specifically when A is differentiable with respect to t and the integral converges, the equation \(\int \frac{d}{dt} (A) dV\) is equal to \(\frac{d}{dt} \int (A) dV\). These conclusions are based on the assumption of a "nice" field and volume V.

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  • Understanding of vector operations, specifically dot products.
  • Knowledge of calculus, particularly differentiation and integration.
  • Familiarity with the concept of differentiability in vector fields.
  • Basic principles of convergence in integrals.
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  • Learn about the conditions for switching differentiation and integration in calculus.
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Hi,
Can someone tell me if: -E dotted with ( A + B )

is equal to -E.A -E.B where E, A and B are all vectors

What I mean is, does the minus sign appear on the E.B bit as well?


Also, is [tex]\int \frac{d}{dt} (A) dV[/tex]

equal to: [tex]\frac{d}{dt} \int (A) dV[/tex]

Thank you
 
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1) Yes off course.

2) Usually the conditions are for all to be "nice": A must be differentiable wrt to t at every point in V, the former integral must converge in order for it to be equal to the latter. I guess you are working with such a "nice" field and volume V, so in that case you may switch differentiation and integration.
 

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