Question

Please answer all parts

uhlqueness!I so, in what way or ways would the proof and the result differ from those given above? IV-25 In the text we defined the gradient in terms of certain partial de- rivatives. It is possible to give an alternative definition similar in form to our definitions of the divergence and the curl. Thus, Here/is a scalar function of position, s a closed surface, and Δν the volume it encloses. As usual, n is a unit vector normal to S and point- ing out from the enclosed volume. (a) Following a procedure similar to the one used in the text in treating the divergence, integrate over a "cuboid" and show that the preceding definition yields the expression (b) Use the alternative definition show that the directional derivative of f in the direstice to by the unit vector û is given by of the gradient given ds Hint: Evaluate over a small cylinder (length As, cross-sectional area ure) whose axis is in the direction of the constant Then divide by the volume of the cylinder (As AA limit as the volume approaches zero.] ΔΑ: see fig (c) Arguing as we did in the text in establishing the divergence theorem, use the alternative definition of the gradient to show that where S is a closed surface enclosing the volume V. (d) Obtain the relation stated in (c) directly from the divergence theorem. [Hint: In JJ,F hs JsS,.F ut / where F êf ê is a constant unit vector.] Verify the relation stated in (c) for the scalar function (e) integrating over the unit cylinder shown in the figure. 17 26 (a C

Answer 1

SoLUTION Given that is a Scala unchon Position Flede encloses Susface auv e Voluwne s ^iven Suation let (x,y,z) be ascalar fid o over the domain > cohere C og

2 业,ê.vf omain Such tha Cos using ch ule, have ナ

6iiven £uation s d Susaceenlosing e 'V S is close Now, we have ·灸

C4 s, 0% m, s) have

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