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On pg 73 of "Tensors, Relativity and Cosmology" The generalized Stokes's theorem in arbitrary $N$-dimensional space is given by: $$\int_c A_mdx^m=\frac{1}{2}\int_S F_{mn}dS^{mn} \tag{1}$$ wher …

On pg 74 of Dalarsson's Tensors, Relativity and Cosmology (The Integral theorems for tensor field chapter), the covariant surface vector was defined as: $$dS_k=\frac{1}{2}\epsilon_{kmn}dx^mdx^n=\frac{ …

In chapter 9.2 of "Tensors, Relativity and Cosmology", the contracted Christoffel symbol of the second kind as a function of the metric tensor was defined as: $$\Gamma_{nm}^m=\frac{1}{2}\left(g^{mk}\f …

In many textbooks & papers the four-velocity vector has the standard definition $$u^{\mu}=\frac{dx^{\mu}}{d\tau}$$ where $\tau$ is the proper time. However, some textbooks define the four-velocity ve …

On Pg 101 of MTW Gravitation, I came across the expression: $$\mathbf{F}(\mathbf{u})=\langle F,u\rangle \tag{1}$$ where $\bf{u}$ is the 4-velocity of the test charged particle, the $\bf{F}$ on the LHS …

When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . …

The proof is often simplified by using the following theorem: "If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g …

Pg 79 of "Tensors, Relativity and Cosmology" In order to construct the geodesic equations which define the curve with a stationary arc length, we may choose the arc length itself as the action int …

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any …

Pg 171 of "Tensors, Relativity and Cosmology" The non-relativistic limit of the metric in a static gravitational field is defined as $$ds^2=\left(1+\frac{2 \phi}{c^2}\right)(dx^0)^2+g_{\alpha \bet …

On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology" For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:$$T^{ …

Consider the vector field $\vec{u}=(xy^2,x^2y,xyz^2)$ The curl of the vector field is $$\nabla \times\vec{u}=(xz^2,-yz^2,0)$$ Consider the line integral of $\vec{u}$ around the ellipse $C$ $x^2+4y^2 …

I'm reading "Textbook in Tensor Calculus and Differential Geometry" by Prasun Kumar Nayak and came across the Weyl tensor/projective curvature tensor $C_{kijl}$. The book states that $$C_{kijl}=R_{kij …

Pg 163 of "Tensors, Relativity and Cosmology" The action integral of a given matter distribution can be written in the form $$I_K=-c\int_\Omega\frac{\rho}{\sqrt{-g}}\frac{ds}{dt}\sqrt{-g} d\Om …

I'm reading Chapter 11 (Normal Modes) of Classical Mechanics (5th ed.) by Berkshire and Kibble and came across this on pg. 253: The kinetic energy in terms in terms of the generalised coordinates is …