拉普拉斯算子，係向量微積分裏面嘅一個算子，佢可以作用喺一個純量函數或者向量函數都得。佢嘅表達式係 ∇ 2 {\displaystyle \nabla ^{2}} 。
對於純量函數，拉普拉斯算子嘅定義係嗰個函數梯度嘅散度，即係 ∇ ⋅ ∇ = ∇ 2 {\displaystyle \nabla \cdot \nabla =\nabla ^{2}} 。作用喺純量函數 f {\displaystyle f} 上面，就係 ∇ ⋅ ∇ f = ∇ 2 f {\displaystyle \nabla \cdot \nabla f=\nabla ^{2}f} 。跟據 ∇ {\displaystyle \nabla } 本身嘅定義，可以得到：
∇ 2 {\displaystyle \nabla ^{2}}
= ∇ ⋅ ∇ {\displaystyle =\nabla \cdot \nabla }
= ( i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z ) ⋅ ( i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z ) {\displaystyle =(\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}})\cdot (\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}})}
= ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 {\displaystyle ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}
所以 ∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 {\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}} 。
至於向量函數 v = v x i + v y j + v z k {\displaystyle \mathbf {v} =v_{x}\mathbf {i} +v_{y}\mathbf {j} +v_{z}\mathbf {k} } ，拉普拉斯算子都可以用同樣嘅方法作用喺佢身上，只不過剩係作用係每個拆出嚟嘅基向量前面嘅函數，即係：
∇ 2 v {\displaystyle \nabla ^{2}\mathbf {v} }
= ∇ 2 v x i + ∇ 2 v y j + ∇ 2 v z k {\displaystyle =\nabla ^{2}v_{x}\mathbf {i} +\nabla ^{2}v_{y}\mathbf {j} +\nabla ^{2}v_{z}\mathbf {k} }
= ( ∂ 2 v x ∂ x 2 + ∂ 2 v x ∂ y 2 + ∂ 2 v x ∂ z 2 ) i + ( ∂ 2 v y ∂ x 2 + ∂ 2 v y ∂ y 2 + ∂ 2 v y ∂ z 2 ) j + ( ∂ 2 v z ∂ x 2 + ∂ 2 v z ∂ y 2 + ∂ 2 v z ∂ z 2 ) k {\displaystyle =({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial z^{2}}})\mathbf {i} +({\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial z^{2}}})\mathbf {j} +({\frac {\partial ^{2}v_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}})\mathbf {k} }
拉普拉斯算子喺唔同科學範疇裏面牽涉到偏微分方程嘅定律就會出現，尤其係物理學。其中一個例子就係波動方程：
∂ 2 u ∂ t 2 = c 2 ∇ 2 u {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}\nabla ^{2}u}
。作用喺純量函數 
上面，就係 
。跟據 
本身嘅定義，可以得到：
。
，拉普拉斯算子都可以用同樣嘅方法作用喺佢身上，只不過剩係作用係每個拆出嚟嘅基向量前面嘅函數，即係：
