James Ebhabha wrote on the scalar triple product of the base vectors. Here is my response:

Please let me have a feedback on this post. Students who want to discuss details can send me their fully typeset comments using the Microsoft equations editor through my email address. I will paste their questions with my response on this page so to benefit other students.

Thanks Sir,

Issue cleared. Will adjust the presentation note accordingly.

Thanks, best regards

one other confusion sir: quoting what you said the day you started tensor algebra “how do you know a tensor? operate ℓ̊†̥ on a vector, if what you get Ȋ̝̊̅§ a vector, then what you have Ȋ̝̊̅§ a tensor” the confusion here Ȋ̝̊̅§ the word ‘operate’ ℓ̊ will appreciate ℓ̊†̥ if the operation can be defined because if ℓ̊ have a vector Ά̲̣̥πϑ ℓ̊ its crosss product ℓ̊ will get a vector ℓ̊†̥ means ℓ̊ will eronously take ℓ̊†̥ for a tensor (ℓ̊ am reserving the word vector for second order tensor Ά̲̣̥πϑ the word tensor for third order tensor). ℓ̊ need †̥o know this because am finding ℓ̊†̥ difficult †̥o find the invariants of tensor with formula (definition) you gave us

There is no confusion at all. First, a cross product is defined as a mapping from a Cartesian product vector space to a vector space. Hence a vector product can only happen between two vectors. Similarly, a dot or scalar product, as defined, can only happen between two vectors. Second. If in a vector product, you take the left vector and the cross product itself together as a joint entity, what you have is called a “Vector Cross”. A

Vector Crossis a tensor. Our definition is consistent!From: ekwonu ikechukwu

Sent: Monday, August 4, 2014 7:03 PM

To: Omotayo Fakinlede

Sir, I want to ask if it is ideal to define the dummy index as the summation index

Sent from my BlackBerry wireless device from MTN

Hello Ekwonu,

The short answer to your question is “yes”. If that does not prove satisfactory, then supply an example and I will probably be able to answer you in greater details.

It is my preference that you send your questions to the s=web page rather than personally to me. It is for us to have participatory discussions even out of class.

OA Fakinlede

Okay sir, but how do ℓ̊ define the word “operate” between a tensor Ά̲̣̥πϑ a vector? Ȋ̝̊̅§ ℓ̊†̥ multiplication, or additon, or dot product or…?

It cannot be addition because addition between a vector and a tensor is not defined. It is a product that occurs in such a way that when the operation of this product is performed by a tensor on a vector, you get a vector. There are textbooks that call this operation a dot product. There are good reasons why Gurtin (our recommended text) does not do that.

A dot product takes place between two vectors and produces a scalar. Many texts call it a scalar product – emphasizing the fact that it results in a scalar.

This operation between a tensor and a vector does NOT produce a scalar so it is a grave mistake to call it a scalar product. Some other authors call it a contraction – a better term than a dot product. Just know it is a kind of product operation that gives out a vector as a result. The scalar product between two tensors is the double dot inner product that we have defined.

Good Afternoon Sir,

Seek clarity on the expansion to obtain the determinant of δ_ijk^rst.

could not get how from the expression : δ_i^k |■(δ_j^r&δ_k^r@δ_j^s&δ_k^s )|-δ_j^k |■(δ_i^r&δ_k^r@δ_i^s&δ_k^s )|+3|■(δ_i^r&δ_j^r@δ_i^s&δ_j^s )|

δ_i^k, -δ_j^k, +3 came about.

Typesetting bad; Referencing, worse. I don’t know what question or example you are referring to.

Sir,

Sorry for the type setting issues. would find an alternative way to type set my questions for clarity.

But, the explanation/ note clears the air properly. really happy with the feedback.

Best regards