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Vec­tor Cal­cu­lus for Elec­tro­mag­net­ism is the back­bone of any com­pre­hens­ive treat­ment of Elec­tro­mag­net­ism. This of course poses chal­lenges to those seek­ing to under­stand the top­ic. None the less, while the vec­tor cal­cu­lus may seem com­plic­ated — it is quite straight for­ward in most cases. In fact, it’s prob­ably fair to say that the math­em­at­ic­al tedi­ous­ness heav­ily out­weighs the actu­al con­cep­tu­al dif­fi­culty. I have attemp­ted to derive (in full) every expres­sion, the­or­em or oth­er­wise which you may require. I hope that this approach will greatly reduce your work­load and sim­pli­fy your stud­ies!Return to Elec­tro­mag­net­ism and Optics

Video # Video Tutori­al Title Remarks
1 Vec­tor Components A quick revi­sion of vec­tor com­pon­ents in order to con­firm the nota­tion I’ll employ in future videos.
2 Scal­ar Dot Product A quick revi­sion dis­cuss­ing both its geo­met­ric mean­ing and its use­ful­ness for cal­cu­lat­ing the flux of a field.
3 a Vec­tor Cross Product 1/2 AxB Cal­cu­la­tion of the vec­tor cross product using matrices.
3 b Vec­tor Cross Product 2/2 AxB The geo­met­ric inter­pret­a­tion of the vec­tor cross product.
4 Law of Cosines  An extremely use­ful trick for ana­lys­ing pairs of vectors.
5 The Sep­ar­a­tion Vector This is the single most import­ant vec­tor in the study of electromagnetism.Every expres­sion requires it as it is the vec­tor dif­fer­ence between the source and detect­or co-ordinates.
6 The ‘Nabla’ Oper­at­or 1/2 ∇ This oper­at­or is required in order to write Maxwell’s Equa­tions in dif­fer­en­tial rather than integ­ral form.
7 The Gradi­ent ∇A The first of the three most use­ful oper­a­tions for ana­lys­ing vec­tor fields.This oper­a­tion is used to ana­lyse rates of change.
8 The Nor­mal Vector A very fun­da­ment­al quant­ity in ana­lys­ing vec­tor fields; the vec­tor per­pen­dic­u­lar to your field.
9 Why the Gradi­ent is Per­pen­dic­u­lar to Functions This video dis­cusses why the gradi­ent of a vec­tor field is ‘nor­mal’ to the field.
10 The Dir­ec­tion­al Derivative This ana­lyses the rate of chance of your vec­tor field in the dir­ec­tion of anoth­er spe­cified vector.
11 The Nabla Oper­at­or 2/2 ∇ Sum­mary of the Nabla oper­at­or and its use for ana­lys­ing vec­tor fields (grad, div, curl).
12 The Diver­gence of a Vec­tor Field ∇⋅A The second of the three most use­ful oper­a­tions for ana­lys­ing vec­tor fields.This oper­a­tion is used to ana­lyse sources or sinks in the field.
13 The Curl of a Vec­tor Field ∇xA The third of the three most use­ful oper­a­tions for ana­lys­ing vec­tor fields.This oper­a­tion is used to ana­lyse rota­tions in the field.
14 Product Rules for Grad/Div/Curl Fun­da­ment­al vec­tor cal­cu­lus for the study of elec­tro­mag­net­ism. The the­ory becomes extremely cum­ber­some without them.
15  Iden­tit­ies for Grad/Div/Curl  Gen­er­al iden­tit­ies for grad, div and curl. I need to make a PDF of this too
16 Product Rule 1 ∇(fA)  Not uploaded
17 Product Rule 2 ∇⋅(fA) The Diver­gence of the product of a scal­ar and vec­tor field.
18 Product Rule 3 ∇x(fA) The Curl of the product of a scal­ar and vec­tor field.
19 Product Rule 4 ∇⋅(AxB) The Diver­gence of the vec­tor field res­ult­ing from a cross product between two vec­tor fields.
20 Product Rule 5 ∇x(AxB) The Curl of the vec­tor field res­ult­ing from a cross product between two vec­tor fields.
21 Product Rule 6 ∇(A⋅B) The Gradi­ent of a scal­ar field res­ult­ing from a scal­ar product between two vec­tor fields.
22 Quo­tient Rule 1  
23 Quo­tient Rule 2  
24 Quo­tient Rule 3  
25 The Lapla­cian Operator  
26 Curl of the Gradient  
27 Diver­gence of the Curl  
28 Curl of the Curl  
29 Fun­da­ment­al The­or­em of Calculus  
30 Fun­da­ment­al The­or­em of Gradients  
31 Green’s Diver­gence Theorem  
32 Stokes’ The­or­em  
33 Integ­ra­tion by Parts Rule 1 A deriv­a­tion of what most people refer to as integ­ra­tion by parts (line integrals)
34 Integ­ra­tion by Parts Example  
35 Integ­ra­tion by Parts Rule 2 For volume and sur­face integrals
36 Integ­ra­tion by Parts Rule 3 For sur­face and line integrals
37 Integ­ra­tion by Parts Rule 4 For volume and sur­face integ­rals involving vec­tor curls
38 Spher­ic­al Polar Co-ords Con­vert­ing rect­an­gu­lar to spher­ic­al co-ordinates
39 Helm­holtz Theorem No deriv­a­tion; just a dis­cus­sion of its importance
40 Dir­ac Delta Func­tion 1/2 Motiv­at­ing the exist­ence of the function/distribution
41 Dir­ac Delta Func­tion 2/2 Illus­trat­ing why its indis­pens­able for physics
42  ∇(1/r) Gradi­ent of one over the Sep­ar­a­tion Vector
43  2(1/r) Lapla­cian of one over the Sep­ar­a­tion Vector
44 Helm­holtz The­or­em 1/2 Deriv­a­tion of the scal­ar potential
45 Helm­holtz The­or­em 2/2 Deriv­a­tion of the vec­tor potential
46 Biot and Savart  Deriv­a­tion of this fun­da­ment­al mag­neto­stat­ic equation
47 Lapla­cian in Spher­ic­al Co-ords   A very pain­ful deriv­a­tion of a most import­ant and fun­da­ment­al operator

vec­tor cal­cu­lus elec­tro­mag­net­ism phys­ics video tutori­als helm­holtz the­or­em scal­ar vec­tor poten­tial voltage vec­tor cal­cu­lus elec­tro­mag­net­ism phys­ics video tutori­als helm­holtz the­or­em scal­ar vec­tor poten­tial voltage vec­tor cal­cu­lus elec­tro­mag­net­ism phys­ics video tutori­als helm­holtz the­or­em scal­ar vec­tor poten­tial voltage 

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