তড়িৎ ও চুম্বক/আলো/ন্যাবলা

একটি স্কেলার ক্ষেত্রের গ্রেডিয়েন্ট, একটি ভেক্টর ক্ষেত্রের ডাইভারজেন্স এবং ঘূর্ণন হলো তিনটি মৌলিক অপারেটর যার মাধ্যমে পদার্থবিজ্ঞানীরা তাদের অধিকাংশ গণনা করেন, বিশেষ করে চল তড়িৎ এবং ফ্লুইড মেকানিক্সে । এই তিনটিই নাবলা $\nabla$ অপারেটর দিয়ে লেখা যায়:

$\nabla =({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})$

$\operatorname {grad} A=\nabla A=({\frac {\partial A}{\partial x}},{\frac {\partial A}{\partial y}},{\frac {\partial A}{\partial z}})$

$\operatorname {div} \mathbf {A} =\nabla \cdot \mathbf {A} ={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}$

$\operatorname {curl} \mathbf {A} =\nabla \times \mathbf {A} =({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}},{\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}},{\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}})$

$\nabla ^{2}=\nabla \cdot \nabla ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$

$\operatorname {curl} \operatorname {curl} \mathbf {A} =\operatorname {grad} \operatorname {div} \mathbf {A} -\nabla ^{2}\mathbf {A}$ তাই নিম্নরূপে লেখা যায়:

$\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$

$\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ এর প্রমাণ:

$(\nabla \times (\nabla \times \mathbf {A} ))_{x}={\frac {\partial }{\partial y}}({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}})-{\frac {\partial }{\partial z}}({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}})$

$={\frac {\partial }{\partial x}}({\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}})-({\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}})A_{x}$

$={\frac {\partial }{\partial x}}({\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}})-({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}})A_{x}$

অন্য দুটি উপাদানের জন্য গণনাও একই রকম।