Product Rules & Amp Laplacian 1
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This document presents proofs of several vector calculus identities involving operators like gradient (?), divergence (??), curl (?¡Á), and Laplacian (?2). It proves the product rule for curl: ?¡Á(¦ÕA)=¦Õ(?¡ÁA)+A¡Á(?¦Õ). It also proves that the curl of a gradient is always zero, and that the curl of curl of a vector A equals the gradient of the divergence of A minus the Laplacian of A. The document defines the Laplacian operator and notes its physical significance for characterizing minima, maxima, and harmonic functions. It concludes with two multiple choice questions about curl and irrotational fields
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For further information about the event please click here.
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Product Rules & Amp Laplacian 1
- 1. PRESENTED BY PRODUCT RULES & LAPLACIAN
- 2. PRODUCT RULES ?. ? ¡Á (??)=?(?¡Á?) + ?¡Á( ??) Proof: ? ¡Á (¦Õ?)=? ¡Á (¦Õ?1i + ¦Õ?2j+¦Õ?3k) = i j k ? ?x ? ?y ? ?z ¦Õ?1 ¦Õ?2 ¦Õ?3
- 3. =[ ? ?y (¦Õ?3) ? ? ?z (¦Õ?2)]i+[ ? ?z (¦Õ?1) ? ?x (¦Õ?3)]j +[ ? ?x ¦Õ?2 ? ? ?y (¦Õ?1)]k =[¦Õ ??3 ?y + ?¦Õ ?? ?3 ? ¦Õ ??2 ?z ? ( ?¦Õ ?? )?2]i+[¦Õ ??1 ?z + ( ?¦Õ ?? )?1 ? ¦Õ ??3 ?x ? ( ?¦Õ ?? )?3]j +[¦Õ ??2 ?x + ?¦Õ ?? ?2 ? ¦Õ ??1 ?y ? ( ?¦Õ ?? )?1]k =? [( ??3 ?y ? ??2 ?y )i+( ??1 ?z ? ??3 ?x )j+( ??2 ?x ? ??1 ?y )k] +[ ?? ?? ?3 ? ?? ?? ?2 i + ?? ?? ?1 ? ?? ?? ?3 + ?? ?? ?2 ? ?? ?? ?1 k]
- 4. =¦Õ ? ¡Á ? + i j k ?¦Õ ?x ?¦Õ ?y ?¦Õ ?z ?1 ?2 ?3 =¦Õ(?¡ÁA) + A¡Á( ?¦Õ)
- 5. ?. ? ¡Á (??)=? Proof: ? ¡Á (?f)=? ¡Á ( ?f ?? ? + ?f ?? j+ ?f ?? ?) = ? ? ? ? ?? ? ?? ? ?? ?f ?? ?f ?? ?f ??
- 6. =[ ? ?? ( ?f ?? ) ? ? ?? ( ?f ?? )]?+[ ? ?? ( ?f ?? ) ? ? ?? ( ?f ?? )]? +[ ? ?? ( ?f ?? ) ? ? ?? ( ?f ?? )]? = 0 ? ¡Á ?? = 0, the reason is that ?? gives a single vector but curl always operate on vector field.
- 7. ?. ? ¡Á ? ¡Á ? = ? ?. ? ? ? ? ? Proof: ? ¡Á ? ¡Á ? = ? ¡Á i j k ? ?x ? ?y ? ?z ?1 ?2 ?3 = ? ¡Á [( ??3 ?y ? ??2 ?y )i+( ??1 ?z ? ??3 ?x )j+( ??2 ?x ? ??1 ?y )k]
- 8. = i j k ? ?x ? ?y ? ?z ??3 ?y ? ??2 ?y ??1 ?z ? ??3 ?x ??2 ?x ? ??1 ?y =[ ? ?? ??2 ?x ? ??1 ?y ? ? ?? ( ??1 ?? ? ??3 ?? )]i+[ ? ?? ??3 ?? ? ??2 ?? ? ? ?? ( ??2 ?x ? ??1 ?y )]j+[ ? ?? ??1 ?? ? ??3 ?? ? ? ?? ( ??3 ?? ? ??2 ?? )]k = (? ?2 ?1 ??2 ? ?2 ?1 ??2 )i+ ? ?2 ?2 ??2 ? ?2 ?2 ??2 j + ? ?2 ?3 ??2 ? ?2 ?3 ??2 ? + ( ?2 ?2 ???? + ?2 ?3 ???? )i+ ?2 ?3 ???? + ?2 ?1 ???? j+ ?2 ?1 ???? + ?2 ?2 ???? k
- 9. = -( ?2 ??2 + ?2 ??2 + ?2 ??2)(?1i+?2j+?3k) + i ? ?x ( ??1 ?x + ??2 ?y + ??3 ?z ) +j ? ?y ( ??1 ?x + ??2 ?y + ??3 ?z ) +k ? ?z ( ??1 ?x + ??2 ?y + ??3 ?z ) = ??2 ? +? ( ??1 ?x + ??2 ?y + ??3 ?z ) =? ?. ? ??2 ? (proved)
- 10. ? ¡Á ? ¡Á ? = ?. ? ? ? ? ?. ? ? ?. ? ? + ? ?. ? Proof: Let A=A1i+A2j+A3k , B=B1i+B2j+B3k A ¡Á B = ? ? ? A1 A2 A3 B1 B2 B3 = (A2 ?3 ? A2 ?3)i + (A1 ?3 ? A3 ?1)j + (A1 ?2 ? A2 ?1)k
- 11. = ? ? ? ? ?? ? ?? ? ?? A2 ?3 ? A2 ?3 A1 ?3 ? A3 ?1 A1 ?2 ? A2 ?1 = i(( ? ?? (A1 ?2 ? A2 ?1) - ? ?? (A1 ?3 ? A3 ?1)) - j( ? ?? (A1 ?2 ? A2 ?1) ? ? ?? (A2 ?3 ? A3 ?2)) +k( ? ?? (A2 ?3 ? A3 ?2)? ? ?? (A3 ?1 ? A1 ?3))
- 12. after solving for i, j , k we will get: Solving for i: = ? ? ?. ? - ?. ? ? ? ? ? ? ?. ? ? ?. ? ? ? Similarly for j and k: = ? ? ?. ? - ?. ? ? ? ? ? ? ?. ? ? ?. ? ? ? = ? ? ?. ? - ?. ? ? ? ? ? ? ?. ? ? ?. ? ? ? Now by adding these three equations ,we will get the desired result i.e. = ?. ? ? ? ? ?. ? ? ?. ? ? + ? ?. ? (proved)
- 13. ? ¡Á ?=0 Proof: Let A=A1i+A2j+A3k ? ¡Á ?= ? ? ? ? ?? ? ?? ? ?? ?1 ?2 ?3 =i ??3 ?? ? ??2 ?? ? ? ??3 ?? ? ??2 ?? + k ??2 ?? ? ??1 ?? =0
- 14. ? ¡Á ?=0 , means field is irrotational.
- 15. LAPLACIAN ? ? ?The laplacian is a differential operator given by the divergence of the gradient of a scalar function V , written as ?2 ? = ?. (??) ? The laplacian of a scalar field is scalar.
- 16. Physical Significance ? As a second derivative, one dimensional laplacian operator is related to minima and maxima. ? when the second derivative is negative(positive), the curvature is concave (convex). ? If the laplacian is zero , the function is harmonic. MCQ¡¯S: 1.The curl of a gradient is a. Zero b. Divergence c. None of these
- 17. 2. If ? ¡Á ? = 0, then the field a. Rotational b. Irrtotational c. solenoidal