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Mixed tensor
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<h2 id="changing-the-tensor-type">Changing the tensor type</h2> <p class="note">Main article: <a href="/facts/Raising_and_lowering_indices/cPsSVpgI">Raising and lowering indices</a></p> <p>Consider the following octet of related tensors: T α β γ , T α β γ , T α β γ , T α β γ , T α β γ , T α β γ , T α β γ , T α β γ . {\displaystyle T_{\alpha \beta \gamma },\ T_{\alpha \beta }{}^{\gamma },\ T_{\alpha }{}^{\beta }{}_{\gamma },\ T_{\alpha }{}^{\beta \gamma },\ T^{\alpha }{}_{\beta \gamma },\ T^{\alpha }{}_{\beta }{}^{\gamma },\ T^{\alpha \beta }{}_{\gamma },\ T^{\alpha \beta \gamma }.} The first one is covariant, the last one contravariant, and the remaining ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor can be lowered using the <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> <i>g</i><i>μν</i>, and a given covariant index can be raised using the inverse metric tensor <i>g</i><i>μν</i>. Thus, <i>g</i><i>μν</i> could be called the <i>index lowering operator</i> and <i>g</i><i>μν</i> the <i>index raising operator</i>. </p><p>Generally, the covariant metric tensor, contracted with a tensor of type (<i>M</i>, <i>N</i>), yields a tensor of type (<i>M</i> − 1, <i>N</i> + 1), whereas its contravariant inverse, contracted with a tensor of type (<i>M</i>, <i>N</i>), yields a tensor of type (<i>M</i> + 1, <i>N</i> − 1). </p> <h3>Examples</h3> <p>As an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3), T α β λ = T α β γ g γ λ , {\displaystyle T_{\alpha \beta }{}^{\lambda }=T_{\alpha \beta \gamma }\,g^{\gamma \lambda },} where T α β λ {\displaystyle T_{\alpha \beta }{}^{\lambda }} is the same tensor as T α β γ {\displaystyle T_{\alpha \beta }{}^{\gamma }} , because T α β λ δ λ γ = T α β γ , {\displaystyle T_{\alpha \beta }{}^{\lambda }\,\delta _{\lambda }{}^{\gamma }=T_{\alpha \beta }{}^{\gamma },} with Kronecker <i>δ</i> acting here like an identity matrix. </p><p>Likewise, T α λ γ = T α β γ g β λ , {\displaystyle T_{\alpha }{}^{\lambda }{}_{\gamma }=T_{\alpha \beta \gamma }\,g^{\beta \lambda },} T α λ ϵ = T α β γ g β λ g γ ϵ , {\displaystyle T_{\alpha }{}^{\lambda \epsilon }=T_{\alpha \beta \gamma }\,g^{\beta \lambda }\,g^{\gamma \epsilon },} T α β γ = g γ λ T α β λ , {\displaystyle T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda }\,T^{\alpha \beta \lambda },} T α λ ϵ = g λ β g ϵ γ T α β γ . {\displaystyle T^{\alpha }{}_{\lambda \epsilon }=g_{\lambda \beta }\,g_{\epsilon \gamma }\,T^{\alpha \beta \gamma }.} </p><p>Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>, g μ λ g λ ν = g μ ν = δ μ ν , {\displaystyle g^{\mu \lambda }\,g_{\lambda \nu }=g^{\mu }{}_{\nu }=\delta ^{\mu }{}_{\nu },} so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">Covariance and contravariance of vectors</a></li> <li><a href="/facts/Einstein_notation/UXwS43wU">Einstein notation</a></li> <li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a></li> <li><a href="/facts/Tensor_(intrinsic_definition)/2KuNm0MK">Tensor (intrinsic definition)</a></li> <li><a href="/facts/Two-point_tensor/uGpBEqkS">Two-point tensor</a></li></ul> <ul><li>D.C. Kay (1988). <i>Tensor Calculus</i>. Schaum’s Outlines, McGraw Hill (USA). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-033484-6.</li> <li>Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973). "§3.5 Working with Tensors". <i><a href="/facts/Gravitation_(book)/1KAfFThq">Gravitation</a></i>. W.H. Freeman & Co. pp. 85–86. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-0344-0.</li> <li>R. Penrose (2007). <i><a href="/facts/The_Road_to_Reality/306chfRy">The Road to Reality</a></i>. Vintage books. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-679-77631-4.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/IndexGymnastics.html">Index Gymnastics</a>, Wolfram Alpha</li></ul>