Prove that the contracted product of two tensors a. b, with components aikbk; is a tensor of order two. b. show that the doubly contracted product aįjbij of a symmetric tensor a and an antisymmetric tensor b vanishes c. show that a symmetric tensor remains symmetric under any transformation of axes d. given a tensor a and a vector b, prove that the contracted products a b and b · a are both vectors, that is show that aijb; and bịajj are components of vectors. note that they are not the same vector unless a is symmetric.

Atank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. the values in the table show the volume v of water remaining in the tank (in gallons) after t minutes.t(min) 5 10 15 20 25 30v(gal) 694 444 250 111 28 0(a) if p is the point (15, 250) on the graph of v, find the slopes of the secant lines pq when q is the point on the graph with t = 5, 10, 20, 25, and 30.(b) estimate the slope of the tangent line at p by averaging the slopes of two secant lines.(c) use a graph of the function to estimate the slope of the tangent line at p. (this slope represents the rate at which the water is flowing from the tank after 15 minutes.)show all steps.

Explain to someone who knows nothing about fossils the study of invertebrate paleontology and how it scientists understand the history of the earth. be sure your explanation contains a minimum of two complete sentences.

Can anybody me with this? ? (picture included.)