$$\nabla_j V^i = \frac\partial V^i\partial x^j + \Gamma^i_jk V^k$$ where $\Gamma^i_jk$ are the Christoffel symbols. In flat space (Cartesian coordinates), the Christoffel symbols vanish, so $\nabla_j V^i = \partial_j V^i$.
If you cannot find the ideal PDF, or you want more variety, consider these free alternatives:
Finding good, free, and comprehensive practice problems can be time-consuming. Here are some of the best, freely available PDF resources focusing on problems and solutions:
Given transformation ( x'^1 = x^1, \quad x'^2 = x^2 + x^3, \quad x'^3 = x^3 ) Find the Jacobian matrix and its determinant.
: A highly detailed resource by Taha Sochi that provides simplified solutions for readers at all levels. You can download it directly from ResearchGate Tensor Algebra and Tensor Analysis for Engineers tensor analysis problems and solutions pdf free
Cover the solution column (or second half of the PDF). Attempt each problem for 10–15 minutes before checking.
: A focused collection of university exam questions covering transformation laws, covariant/contravariant ranks, and metric tensors. Tensor Notation Guide (UBC)
Avoid sites asking for credit card information or promising “instant download” after a survey. Stick to .edu , .org , or reputable academic sharing platforms.
gij;k=𝜕gij𝜕xk−[ik,j]−[jk,i]g sub i j ; k end-sub equals the fraction with numerator partial g sub i j end-sub and denominator partial x to the k-th power end-fraction minus open bracket i k comma j close bracket minus open bracket j k comma i close bracket $$\nabla_j V^i = \frac\partial V^i\partial x^j + \Gamma^i_jk
B1=g11B1+g12B2=(2)(5)+(0)(-1)=10cap B sub 1 equals g sub 11 cap B to the first power plus g sub 12 cap B squared equals open paren 2 close paren open paren 5 close paren plus open paren 0 close paren open paren negative 1 close paren equals 10 Expand for B2cap B sub 2
Prove: ( \varepsilon_ijk \varepsilon_imn = \delta_jm\delta_kn - \delta_jn\delta_km )
( \delta_ii = \delta_11+\delta_22+\delta_33 = 1+1+1 = 3 ) (in 3D).
(Generated for open educational use) License: CC BY-NC-SA 4.0 Here are some of the best, freely available
Prove the "Quotient Rule" for tensors: If $A_i$ is known to be a covariant vector and the relation $B^ijA_j = C^i$ holds for arbitrary $A_j$, and $C^i$ transforms as a contravariant vector, prove that $B^ij$ is a contravariant tensor of rank 2.
If abstract index notation feels confusing, write out the components explicitly for a 2D or 3D space to build spatial intuition.
Some advanced free PDFs include metric-specific exercises.
δ̄km=𝜕x̄m𝜕xi𝜕xj𝜕x̄kδjidelta bar sub k to the m-th power equals the fraction with numerator partial x bar to the m-th power and denominator partial x to the i-th power end-fraction the fraction with numerator partial x to the j-th power and denominator partial x bar to the k-th power end-fraction delta sub j to the i-th power