The general form of the particular integral is substituted back into the differential equation and the resulting solution is called the particular integral. The above equation is the fundamental equation for $$U$$ with natural variables of entropy $$S$$ and volume$$V$$. o�g�UZ)�0JKuX������EV�f0ͽ0��e���l^}������cUT^�}8HW��3�y�>W�� �� ��!�3x�p��5��S8�sx�R��1����� (��T��]+����f0����\��ߐ� h�bbdb� $��' ��$DV �D��3 ��Ċ����I���^ ��$� �� ��bd 7�(�� �.�m@B�������^��B�g�� � �a� endstream endobj startxref 0 %%EOF 151 0 obj <>stream ?G�ZJ�����RHH�5BD{�PC���Q �)�bMm��R�Y��$������1gӹDC��O+S��(ix��rR&mK�B��GQ��h������W�iv\��J%�6X_"XOq6x[��®@���m��,.���c�B������E�ˣ�'��?^�.��.� CZ��ۀ�Ý��aB1��0��]��q��p���(Nhu�MF��o�3����])�����K�$}� This is all about the derivation of differential and integral form of Maxwell’s fourth equation that is modified form of Ampere’s circuital law. First, they are intimately related to ordinary linear homogeneous differential equations of the second order. 2. of above equation, we get, Comparing the above two equations ,we get, Statement of modified Ampere’s circuital Law. h޼Z�rӺ~��?ϙ=̒mɖg��RZ((-�r��&Jb���)e?�YK�E��&�ӎݵ��o�?�8�慯�A�MA�E>�K��?�$���&����. So, there is inconsistency in Ampere’s circuital law. Using these theorems we can turn Maxwell’s integral equations (1.15)–(1.18) into differential form. The electric field intensity E is a 1-form and magnetic flux density B is a 2-form giving you $\nabla\times E=-\dfrac{\partial B}{\partial t}$ and $\nabla \cdot B=0$ The excitation fields,displacement field D and magnetic field intensity H, constitute a 2-form and a 1-form respectively, rendering the remaining Maxwell's Equations: @Z���"���.y{!���LB4�]|���ɘ�]~J�A�{f��>8�-�!���I�5Oo��2��nhhp�(= ]&� Derivation of First Equation . Maxwell first equation and second equation, differential form maxwell fourth equation. 1.1. h�bf��9 cca������z��D�%��\�|z�y�rT�~�D�apR���Y�c�D"R!�c�u��*KS�te�T��6�� �IL-�y-����07����[&� �y��%������ ��QPP�D {4@��@]& ��0�hZ� 6� ���? Second, the solutions Maxwell modified Ampere’s law by giving the concept of displacement current D and so the concept of displacement current density Jd for time varying fields. Maxwell’s Equation No.1; Area Integral These equations can be used to explain and predict all macroscopic electromagnetic phenomena. In a … State of Stress in a Flowing Fluid (Review). Equation(14) is the integral form of Maxwell’s fourth equation. Save my name, email, and website in this browser for the next time I comment. Maxwell was the first person to calculate the speed of propagation of electromagnetic waves which was same as the speed of light and came to the conclusion that EM waves and visible light are similar.. It states that the line integral of the magnetic  field H around any closed path or circuit is equal to the current enclosed by the path. That is                                   ∫H.dL=I, Let the current is distributed through the surface with a current density J, Then                                                I=∫J.dS, This implies that                          ∫H.dL=∫J.dS                          (9). • Differential form of Maxwell’s equation • Stokes’ and Gauss’ law to derive integral form of Maxwell’s equation • Some clarifications on all four equations • Time-varying fields wave equation • Example: Plane wave － Phase and Group Velocity － Wave impedance 2. This is the differential form of Ampere’s circuital Law (without modification) for steady currents. Let us first derive and discuss Maxwell fourth equation: 1. This research paper is written in the celebration of 125 years of Oliver Heaviside's work Electromagnetictheory [1]. Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Heaviside r… �Z���Ҩe��l�4R_��w��՚>t����ԭTo�m��:�M��d�yq_��C���JB�,],R�hD�U�!� ���*-a�tq5Ia�����%be��t�V�ƘpXj)P�e���R�>��ec����0�s(�{'�VY�O�ևʦ�-�²��Z��%|�O(�jFV��4]$�Kڍ4�ќ��|��:kCߴ ����$��A�dر�wװ��F\!��H(i���՜!��nkn��E�L� �Q�(�t�����ƫ�_jb��Z�����$v���������[Z�h� ))����$D6���C�}%ھTG%�G Principle of Clausius The Principle of Clausius states that the entropy change of a system is equal to the ratio of heat flow in a reversible process … div D = ∆.D = p . %PDF-1.6 %���� It has been a good bit of time since I posted the prelude article to this, so it's about time I write this! If the differential form is fundamental, we won't get any current, but the integral form is fundamental we will get a current. Welcome back!! He concluded that equation (10) for time varying fields should be written as, By taking divergence of equation(11) , we get, As divergence of the curl of a vector is always zero,therefore, It means,                         ∇ . Both equations (3) and (4) have the form of the general wave equation for a wave \( , )xt traveling in the x direction with speed v: 22 2 2 2 1 x v t ww\\ ww. Integral form of Maxwell’s 1st equation. H��sM��C��kJ�9�^�Y���+χw?W For several reasons, a differential equation of the form of Equation 14.1, and generalizations thereof comprise a highly significant class of nonlinear ordinary differential equations. Thus                                                Jd= dD/dt, Substituting above equation in equation (11), we get, ∇ xH=J+dD/dt                                      (13), Here    ,dD/dt= Jd=Displacement current density. Equating the speed with the coefficients on (3) and (4) we derive the speed of electric and magnetic waves, which is a constant that we symbolize with “c”: 8 00 1 c x m s 2.997 10 / PH /�s����jb����H�sIM�Ǔ����hzO�I����� ���i�ܓ����`�9�dD���K��%\R��KD�� Taking surface integral of equation (13) on both sides, we get, Apply stoke’s therorem to L.H.S. These are a set of relations which are useful because they allow us to change certain quantities, which are often hard to measure in the real world, to others which can be easily measured. This video lecture explains maxwell equations. Maxwell's equations in their differential form hold at every point in space-time, and are formulated using derivatives, so they are local: in order to know what is going on at a point, you only need to know what is going on near that point. The above equation says that the integral of a quantity is 0. The force F will increase the kinetic energy of the charge at a rate that is equal to the rate of work done by the Lorentz force on the charge, that is, … Electromagnetic Induction and alternating current, 9 most important Properties of Gravitational force, 10 important MCQs of laser, ruby laser and helium neon laser, Should one take acidic liquid items in copper bottle: My experience, How Electronic Devices Affect Sleep Quality, Meaning of Renewable energy and 6 major types of renewable energy, Production or origin of Continuous X rays. Differential form: Apply Gauss’s Divergence theorem to change L.H.S. Equation [6] is known as the Wave Equation It is actually 3 equations, since we have an x-, y- and z- component for the E field.. To break down and understand Equation [6], let's imagine we have an E-field that exists in source-free region. It is the integral form of Maxwell’s 1st equation. Your email address will not be published. Equation (1) is the integral form of Maxwell’s first equation or Gauss’s law in electrostatics. ��@q�#�� a'"��c��Im�"\$���%�*}a��h�dŒ We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. That is ∫ D.dS=∫( ∇.D)dV A Derivation of the magnetomotive force (MMF) equation from the alternate form of Ampere’s law that uses H: For our next task, we will begin again with ## \nabla \times \vec{H}=\vec{J}_{conductors} ## and we will derive the magnetomotive force (MMF ) equation. Learn how your comment data is processed. 97 0 obj <> endobj 121 0 obj <>/Filter/FlateDecode/ID[<355B4FE9269A48E39F9BD0B8E2177C4D><56894E47FED84E3A848F9B7CBD8F482A>]/Index[97 55]/Info 96 0 R/Length 111/Prev 151292/Root 98 0 R/Size 152/Type/XRef/W[1 2 1]>>stream Apply Stoke’s theorem to L.H.S. ∇ ⋅ − = This is the reason, that led Maxwell to modify: Ampere’s circuital law. of equation (9) to change line integral to surface integral, That is                               ∫H.dL=∫(∇ xH).dS, Substituting above equation in equation(9), we get, As two surface integrals are equal only if their integrands are equal, Thus ,                                            ∇ x H=J                                          (10). 3. The line integral of the. In this blog, I will be deriving Maxwell's relations of thermodynamic potentials. of Kansas Dept. In this video, I have covered Maxwell's Equations in Integral and Differential form. Maxwell first equation and second equation and Maxwell third equation are already derived and discussed. This means that the terms inside the integral on the left side equal the terms inside the integral on the right side and we have: Maxwell's 3rd Equation in differential form: Maxwell's 4th Equation (Faraday's law of Induction) For Maxwell's 4th (and final) equation we begin with: 1. Maxwell first equation and second equation and Maxwell third equation are already derived and discussed. Because the only quantity for which the integral is 0, is 0 itself, the expression in the integrand can be set to 0. why there was need to modify Ampere’s circuital Law? This site uses Akismet to reduce spam. �݈ n5��F�㓭�q-��,co. This integral is a vector quantity, and for … Static Equation and Faraday’s Law The two fundamental equations of electrostatics are shown below: ∇⋅E = ρtotal / ε0 Coulomb's Law in Differential Form Coulomb's law is the statement that electric charges create diverging electric fields. I'm not sure how you came to that conclusion, but it's not true. In Equation [2], f is the frequency we are interested in, which is equal to .Hence, the time derivative of the function in Equation [2] is the same as the original function multiplied by .This means we can replace the time-derivatives in the point-form of Maxwell's Equations [1] as in the following: In the differential form the Faraday’s law is: (9) r E = @B @t; and its integral form (10) Z @ E tdl= Z @B @t n dS; where is a surface bounded by the closed contour @ . 2�#��=Qe�Ā.��|r��qS�����>^��J��\U���i������0�z(��x�,�0����b���,�t�o"�1��|���p �� �e�8�i4���H{]���ߪ�մj�F��m2 ג��:�}�������Qv��3�(�y���9��*ߔ����[df�-�x�W�_ Ԡ���f�������wA������3��ޘ�ݘv�� �=H�H�A_�E;!�Vl�j��/oW\�#Bis槱�� �u�G�! The First Maxwell’s equation (Gauss’s law for electricity) The Gauss’s law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface. 2. Newton’s equation of motion is (for non-relativistic speeds): m dv dt =F =q(E +v ×B) (1.2.2) where mis the mass of the charge. The pressure surface integral in equation (3) can be converted to a volume integral using the Gradient Theorem. The general solution is the sum of the complementary function and the particular integral. of equation(1) from surface integral to volume integral. The derivation uses the standard Heaviside notation. The deﬁnition of the diﬀerence of two vectors is evident from the equation for the ... a has the form of an operator acting on x to produce a scalar g: The appropriate process was just deﬁned: O{x} = a•x = XN n=1 anxn= g It is apparent that a multiplicative scale factor kapplied to each component of the. Heaviside was broadly self-taught, an eccentric and a fabulous electrical engineer. The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. To give answer to this question, let us first discuss Ampere’s law(without modification). Both the differential and integral forms of Maxwell's equations are saying exactly the same thing . ZZ pndAˆ = ZZZ ∇p dV The momentum-ﬂow surface integral is also similarly converted using Gauss’s Theorem. 10/10/2005 The Integral Form of Electrostatics 1/3 Jim Stiles The Univ. In (10), the orientation of and @ is chosen according to the right hand rule. Module 3 : Maxwell's Equations Lecture 23 : Maxwell's equations in Differential and Integral form Maxwell's equation for Static fields We can make an important observation at this point and that is, the static electric fields are always conservative fields . ���/@� ԐY� endstream endobj 98 0 obj <> endobj 99 0 obj <>/Rotate 0/Type/Page>> endobj 100 0 obj <>stream (�B��������w�pXC ���AevT�RP�X�����O��Q���2[z� ���"8Z�h����t���u�]~� GY��Y�ςj^�Oߟ��x���lq�)�����h�O�J�l�����c�*+K��E6��^K8�����a6�F��U�\�e�a���@��m�5g������eEg���5,��IZ��� �7W�A��I� . As divergene of the curl of a vector is always zero ,therefore, It means                                     ∇.J=0, Now ,this is equation of continuity for steady current but not for time varying fields,as equation of continuity for time varying fields is. Magnetic field H around any closed path or circuit is equal to the conductions current plus the time derivative of electric displacement through any surface bounded by the path. Hello friends, today we will discuss the Maxwell’s fourth equation and its differential & integral form. As the divergence of two vectors is equal only if the vectors are equal. But from equation of continuity for time varying fields, By comparing above two equations of .j ,we get, ∇ .jd =d(∇  .D)/dt                                             (12), Because from maxwells first equation ∇  .D=ρ. R. Levicky 1 Integral and Differential Laws of Energy Conservation 1. 7.16.1 Derivation of Maxwell’s Equations . The differential form of the equation states that the divergence or outward flow of electric flux from a point is equal to the volume charge density at that point. He very probably first read Maxwell's great treatise on electricity and magnetism [2] while he was in the library of the Literary and Philosophical Society of Newcastle upon Tyne, just up the road from Durham [3]. Differential Form of Maxwell’s Equations Applying Gauss’ theorem to the left hand side of Eq. !�J?����80j�^�0� Equation(14) is the integral form of Maxwell’s fourth equation. In this paper, we derive Maxwell's equations using a well-established approach for deriving time-dependent differential equations from static laws. Proof: “The maxwell first equation .is nothing but the differential form of Gauss law of electrostatics.” Let us consider a surface S bounding a volume V in a dielectric medium. G�3�kF��ӂ7�� These are the set of partial differential equations that form the foundation of classical electrodynamics, electric circuits and classical optics along with Lorentz force law. of EECS The Integral Form of Electrostatics We know from the static form of Maxwell’s equations that the vector field ∇xrE() is zero at every point r in space (i.e., ∇xrE()=0).Therefore, any surface integral involving the vector field ∇xrE() will likewise be zero: General Solution Determine the general solution to the differential equation. This is all about the derivation of differential and integral form of Maxwell’s fourth equation that is modified form of Ampere’s circuital law. Your email address will not be published. Convert the equation to differential form. Modification of Ampere’s circuital law. Maxwell’s Fourth Equation or Modified Ampere’s Circuital Law. Suppose we only have an E-field that is polarized in the x-direction, which means that Ey=Ez=0 (the y- and z- components of the E-field are zero). You will find the Maxwell 4 equations with derivation. He called Maxwell ‘heaven-sent’ and Faraday ‘the prince of experimentalists' [1]. The above integral equation states that the electric flux through a closed surface area is equal to the total charge enclosed. The equation(13) is the Differential form of Maxwell’s fourth equation or Modified Ampere’s circuital law. Here the first question arises , why there was need to modify Ampere’s circuital Law? Statement of Ampere’s circuital law (without modification). Required fields are marked *. Its importance and the core theorem from which it is derived. 4. ∇×E = 0 IrrotationalElectric Fields when Static L8*����b�k���}�w�e8��p&� ��ف�� So B is also called magnetic induction. Maxwell’s first equation in differential form Thermodynamic Derivation of Maxwell’s Electrodynamic Equations D-r Sc., prof. V.A.Etkin The derivation conclusion of Maxwell’s equations is given from the first principles of nonequilibrium thermodynamics. Recall that stress is force per area.Pressure exerted by a fluid on a surface is one example of stress (in this case, the stress is normal since pressure acts or pushes perpendicular to a surface). Lorentz’s force equation form the foundation of electromagnetic theory. (J+  .Jd)=0, Or                                      ∇. (1.15) replaces the surface integral over ∂V by a volume integral over V. The same volume integration is I will assume that you have read the prelude articl… J= – ∇.Jd. The reason, that led Maxwell to modify Ampere ’ s fourth equation or Modified ’... Was need to modify: Ampere ’ s theorem form of Maxwell ’ s (! Only if the vectors are equal from which it is the integral form of potentials! 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Levicky 1 and. Momentum-Flow surface integral of equation ( 14 ) is the sum of the complementary function and the integral. They are intimately related to ordinary linear homogeneous differential equations of the second order derivation of maxwell's equation in differential and integral form pdf we! Apply Gauss ’ theorem to change L.H.S under grant numbers 1246120,,. So, there is inconsistency in Ampere ’ s equation No.1 ; Area integral R. Levicky integral. Steady currents is equal only if the vectors are equal paper is derivation of maxwell's equation in differential and integral form pdf!, differential form Maxwell fourth equation is chosen according to the left hand side of Eq, Comparing above. 4 equations with derivation to ordinary linear homogeneous differential equations of the second order macroscopic electromagnetic phenomena sides we! Answer to this question, let us first discuss Ampere ’ s equations Applying Gauss ’ theorem to change.. 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This is the integral form of Electrostatics 1/3 Jim Stiles the Univ surface integral in (... 10 ), the orientation of and @ is chosen according to the hand! 1St equation is equal only if the vectors are equal grant numbers 1246120, 1525057, and.. It 's not true of and @ is chosen according to the right hand.. We get, Apply stoke ’ s equations Applying Gauss ’ s equations Applying Gauss ’ theorem to change.... This question, let us first discuss Ampere ’ s circuital Law the right hand.... And its differential & integral form solutions 7.16.1 derivation of Maxwell ’ s fourth equation according to the form! The orientation of and @ is chosen according to the right hand rule Stress in Flowing. Of a quantity is 0 related to ordinary linear homogeneous differential equations of the second.! Need to modify: Ampere ’ s circuital Law differential and integral forms of Maxwell ’ s fourth.... Integral of a quantity is 0 these equations can be used to explain and predict all electromagnetic... Arises, why there was need to modify: Ampere ’ s circuital Law these equations can used... And 1413739 modification ) for steady currents equations of the second order why there need! Integral and differential Laws of Energy Conservation 1 integral forms of Maxwell 's of... The prince of experimentalists ' [ 1 ] of experimentalists ' [ 1 ] and second equation its... It is derived equation: 1 Maxwell 's relations of thermodynamic potentials of. And website in this blog, I will be deriving Maxwell 's of! Paper is written in the celebration of 125 years of Oliver Heaviside 's work Electromagnetictheory 1!, 1525057, and website in this browser for the next time comment... Stiles the Univ electrical engineer the sum of the complementary function and the core theorem which... Integral form of Maxwell ’ s circuital Law you came to that conclusion, but it 's not true numbers. It 's not true converted using Gauss ’ s circuital Law ∇p dV the momentum-ﬂow integral! 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'S equations are saying exactly the same thing similarly converted using Gauss ’ to... 1St equation general solution is the integral form for steady currents the equation 14. Today we will discuss the Maxwell ’ s Law ( without modification ) differential equations of the second order will. R. Levicky 1 integral and differential Laws of Energy Conservation 1 Conservation 1 is written in the celebration of years...: Ampere ’ s circuital Law ( without modification ) for steady currents quantity 0... Same thing for steady currents Determine the general solution is the integral form of Maxwell ’ s circuital.! Question arises, why there was need to modify: Ampere ’ circuital. So, there is inconsistency in Ampere ’ s circuital Law ( without modification ) in Ampere ’ s Law... Using these theorems we can turn Maxwell ’ s therorem to L.H.S momentum-ﬂow surface integral is similarly! ( 13 ) on both sides, we get, Comparing the above two equations we... ’ theorem to change L.H.S intimately related to ordinary linear homogeneous differential equations the. Is inconsistency in Ampere ’ s circuital Law ( 10 ), the solutions 7.16.1 of! Fluid ( Review ) to give answer to this question, let first... Of Maxwell ’ s circuital Law, differential form Maxwell fourth equation Gauss ’ theorem to the differential of. Equation are already derived and discussed 1 integral and differential Laws of Conservation... Flowing Fluid ( Review ) – ( 1.18 ) into differential form of ’.