mail  mail@stemandmusic.in

    

call  +91-9818088802

Rules for Calculating Derivatives and Differentials of Vector Functions

1. A Vector Function is a Vector which has atleast One of its Components as a Function of One or More Variables.
2. Calculating Derivative or Partial Derivative of a Vector Function with respect to any Variable involves Calculating the Derivative or Partial Derivative of All its Components with respect to that Variable.
   
   Calculating Differential of a Vector Function of a Single Variable involves Calculating its Derivative with respect to that Variable and Multiplying it with the Differential of that Variable.
   
   Calculating Partial Differential of a Vector Function of Multiple Variables with respect to Any One Variable involves Calculating its Derivative with respect to that Variable and Multiplying it with the Differential of that Variable.
   
   Calculating Total Differential of a Vector Function of Multiple Variables involves Adding the Partial Differentials of a Vector Function with respec to all its Variables.
   
   Please note that All these Calculation and Representation of Derivatives / Partial Derivatives / Differentials / Partial Differentials / Total Differentials are done as specified in the Rules for Calculating Derivatives and Differentials for Functions of a Single Variable and Rules for Calculating Derivatives and Differentials for Functions of Multiple Variables.
3. If \(\vec{A}\) and \(\vec{B}\) are Vector Functions of Same Set of Variables, the Derivative of their Dot Product and Cross Product with respect to Any Variable are given as
   
   \((\vec{A} \cdot \vec{B})' = (\vec{A}\hspace{1mm}' \cdot \vec{B}) + (\vec{A} \cdot \vec{B}\hspace{1mm}')\)
   
   \((\vec{A} \times \vec{B})' = (\vec{A}\hspace{1mm}' \times \vec{B}) + (\vec{A} \times \vec{B}\hspace{1mm}')\)
4. If \(S\) is Scalar Function and \(\vec{A}\) is a Vector Function of Same Same Set of Variables, the Derivative of \(\vec{A}\) Scaled by \(S\) with respect to Any Variable is given as
   
   \((S\vec{A})' = (S\hspace{1mm}'\vec{A}) + (S\vec{A}\hspace{1mm}')\)
5. For any Vector Function \(\vec{R}\) and Any Constant \(n\)
   
   \(((\vec{R} \cdot \vec{R})^n)' = 2n (\vec{R} \cdot \vec{R})^{n-1} (\vec{R} \cdot \vec{R}\hspace{1mm}') \)
   
   \(|\vec{R}|' = (\sqrt{\vec{R} \cdot \vec{R}})' = {\Large \frac{\vec{R} \cdot \vec{R}\hspace{1mm}'}{|\vec{R}|}}\)
   
   \((|\vec{R}|^n)' = {\Large \frac{ n|\vec{R}|^{n-1}(\vec{R} \cdot \vec{R}\hspace{1mm}')}{|\vec{R}|}}= n|\vec{R}|^{n-2}(\vec{R} \cdot \vec{R}\hspace{1mm}')\)
   
   \(((\vec{R} \times \vec{R}\hspace{1mm}')^n)' = n (\vec{R} \times \vec{R}\hspace{1mm}')^{n-1}(\vec{R} \times \vec{R}\hspace{1mm}'') \)
   
   \(|\vec{R} \times \vec{R}\hspace{1mm}'|' = {\sqrt{(\vec{R} \times \vec{R}\hspace{1mm}') \cdot (\vec{R} \times \vec{R}\hspace{1mm}')}}^{\hspace{2mm}'} = {\Large \frac{(\vec{R} \times \vec{R}\hspace{1mm}') \cdot (\vec{R} \times \vec{R}\hspace{1mm}'')}{|\vec{R} \times \vec{R}\hspace{1mm}'|}}\)
   
   \((|\vec{R} \times \vec{R}\hspace{1mm}'|^n)' = {\Large \frac{ n|\vec{R} \times \vec{R}\hspace{1mm}'|^{n-1}((\vec{R} \times \vec{R}\hspace{1mm}') \cdot (\vec{R} \times \vec{R}\hspace{1mm}''))}{|\vec{R} \times \vec{R}\hspace{1mm}'|}}= n|\vec{R} \times \vec{R}\hspace{1mm}'|^{n-2}((\vec{R} \times \vec{R}\hspace{1mm}') \cdot (\vec{R} \times \vec{R}\hspace{1mm}''))\)