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( ∂/∂x , dx∧dy ) = dy and ( ∂/∂x , dy∧dx ) = -dy

( ∂/∂z , dx∧dy∧dz ) = dx∧dy and ( ∂/∂z , dx∧dz∧dy ) = -dx∧dy

Have I got this right ? Thanks

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In summary, the conversation discusses the use of interior products of vectors and differential forms in differential geometry, and how to manipulate them to get the desired result. The speaker also confirms that their understanding is correct, and the expert provides a method for verifying the identity using the fact that dx∧dy=dx⊗dy−dy⊗dx.

- #1

- 770

- 61

( ∂/∂x , dx∧dy ) = dy and ( ∂/∂x , dy∧dx ) = -dy

( ∂/∂z , dx∧dy∧dz ) = dx∧dy and ( ∂/∂z , dx∧dz∧dy ) = -dx∧dy

Have I got this right ? Thanks

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You can then use the fact that ##dx \wedge dy = dx \otimes dy - dy \otimes dx## to verify your identity. When you have ##\partial/\partial x## you would get

$$

\left(\frac{\partial}{\partial x}, dy \wedge dx\right) = \underbrace{dy(\partial_x)}_{=0} dx - \underbrace{dx(\partial_x)}_{=1} dy = -dy.

$$

Of course, when you have the differential form expressed in the coordinate differentials, you can generally just anti-commute the relevant coordinate differential to the first position. All terms where it is not in the first position will vanish identically and the interior product will just correspond to anti-commuting the relevant coordinate differential to the front and then removing it from the exterior product.

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An interior product with differential forms is a mathematical operation that combines a vector field and a differential form to produce a new differential form. It is denoted by the symbol "<i>" and is also known as the inner product or contraction.

The interior product is defined as the contraction of a vector and a differential form, resulting in a new differential form. It is defined using the exterior product and the Hodge star operator.

The interior product follows several important properties, such as linearity, associativity, and compatibility with the exterior derivative. It also satisfies the Leibniz rule, which states that the interior product of a differential form with a vector field is equal to the exterior derivative of the interior product of the differential form with the vector field.

The interior product plays a crucial role in differential geometry, particularly in the study of manifolds. It allows for the definition of important geometric concepts such as Lie derivatives, Lie brackets, and Lie algebras. It also helps in the formulation of Maxwell's equations in electromagnetism and the study of vector calculus.

The interior product has various applications in physics, engineering, and other fields. It is used in the study of fluid dynamics, electromagnetism, and general relativity. It is also used in computer graphics and computer vision to model and manipulate geometric objects. In addition, the interior product is used in optimization problems and machine learning algorithms.

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