Scalar Product of a diffrential.

In summary, the scalar product of a differential is a mathematical operation that involves multiplying two differentials or infinitesimal quantities together, also known as the dot product or inner product. It is calculated by multiplying the magnitudes of the two differentials and the cosine of the angle between them. The purpose of this operation is to aid in calculations in various fields of mathematics and physics, such as vector calculus, mechanics, and electromagnetism. It is not commutative, meaning the order of multiplication affects the result. Additionally, the scalar product of a differential can be negative, which occurs when the angle between the two differentials is greater than 90 degrees.
  • #1
raisin_raisin
27
0
Hey, in my textbook they keep doing this and I can't follow
for example r.[tex]\ddot{}r[/tex] = 1/2 [tex]\ddot{}r^{}^2{}[/tex]
and r.[tex]\dot{}r[/tex] = 1/2 [tex]\dot{}r^2{}[/tex].

Can anyone explain this to me? I know I should probably know it.

P.S Can't quite get the dot product to look right apologies.
 
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  • #2
(r²)' = 2rr' by chain rule. Or if the r's are supposed to be vector functions:

(r.r)'
= r'.r + r.r' by product rule for dot product
= r.r' + r.r' by commutativity of dot product
= 2r.r'
 

1. What is the definition of Scalar Product of a Differential?

The scalar product of a differential is a mathematical operation that involves multiplying two differentials or infinitesimal quantities together. It is also known as the dot product or inner product.

2. How is Scalar Product of a Differential calculated?

The scalar product of two differentials, dx and dy, is calculated by multiplying their magnitudes and the cosine of the angle between them. In mathematical notation, it is represented as dx * dy = |dx| * |dy| * cosθ.

3. What is the purpose of Scalar Product of a Differential?

The scalar product of a differential is useful in many areas of mathematics and physics, such as vector calculus, mechanics, and electromagnetism. It allows us to calculate the work done by a force, the angle between two vectors, and the projection of one vector onto another.

4. Is Scalar Product of a Differential commutative?

No, the scalar product of a differential is not commutative. This means that the order in which the differentials are multiplied affects the result. In other words, dx * dy is not equal to dy * dx.

5. Can Scalar Product of a Differential be negative?

Yes, the scalar product of a differential can be negative. This occurs when the angle between the two differentials is greater than 90 degrees, resulting in a negative value for the cosine. This can also be interpreted as the two differentials being in opposite directions.

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