Laplace operator

The Laplace operator is a mathematical operator first introduced by Pierre-Simon Laplace . It is a linear differential operator within multidimensional analysis . It is usually notated by the symbol , the capital letter Delta of the Greek alphabet . ${\ displaystyle \ Delta}$

The Laplace operator appears in many differential equations that describe the behavior of physical fields . Examples are the Poisson equation for electrostatics , the Navier-Stokes equations for flows of liquids or gases and the diffusion equation for heat conduction .

definition

The Laplace operator assigns the divergence of its gradient to a twice differentiable scalar field , ${\ displaystyle f}$

${\ displaystyle \ Delta f = \ operatorname {div} \ left (\ operatorname {grad} \, f \ right),}$

or notated with the Nabla operator

${\ displaystyle \ Delta f = \ nabla \ cdot (\ nabla f) = (\ nabla \ cdot \ nabla) f = \ nabla ^ {2} f.}$

The formal “ scalar product ” of the Nabla operator with itself results in the Laplace operator. The notation for the Laplace operator is often found in English-speaking countries in particular . ${\ displaystyle \ nabla ^ {2}}$

Since the divergence operator and the gradient operator are independent of the selected coordinate system , the Laplace operator is also independent of the selected coordinate system. The representation of the Laplace operator in other coordinate systems results from the coordinate transformation using the chain rule . ${\ displaystyle \ operatorname {div}}$${\ displaystyle \ operatorname {grad}}$

In -dimensional Euclidean space results in Cartesian coordinates${\ displaystyle n}$

${\ displaystyle \ Delta f = \ sum _ {k = 1} ^ {n} {\ partial ^ {2} f \ over \ partial x_ {k} ^ {2}}.}$

In one dimension, the Laplace operator is reduced to the second derivative:

${\ displaystyle \ Delta f = f ''}$

The Laplace operator of a function can also be represented as a trace of its Hessian matrix :

${\ displaystyle \ Delta f = \ mathrm {Spur} (H (f))}$

The Laplace operator can also be applied to vector fields . With the dyadic product " " becomes with the Nabla operator${\ displaystyle \ otimes}$ ${\ displaystyle \ nabla}$

${\ displaystyle \ Delta {\ vec {v}}: = (\ nabla \ cdot \ nabla) {\ vec {v}} = \ nabla \ cdot (\ nabla \ otimes {\ vec {v}}) = \ operatorname {div (grad} ({\ vec {v}}) ^ {\ top})}$

Are defined. The superscript stands for transposition . In the literature there is also a divergence operator that transposes its argument accordingly . With this operator, analogous to the scalar field, we write ${\ displaystyle {} ^ {\ top}}$${\ displaystyle \ operatorname {\ widetilde {div}} T = \ operatorname {div} (T ^ {\ top})}$

${\ displaystyle \ Delta {\ vec {v}} = \ operatorname {{\ widetilde {div}} (grad} \, {\ vec {v}})}$

The rotation operator is especially true in three dimensions ${\ displaystyle \ operatorname {red}}$

${\ displaystyle \ Delta {\ vec {v}} = (\ nabla \ cdot \ nabla) {\ vec {v}} = \ nabla (\ nabla \ cdot {\ vec {v}}) - \ nabla \ times ( \ nabla \ times {\ vec {v}}) = \ operatorname {grad (div} ({\ vec {v}})) - \ operatorname {rot (rot} ({\ vec {v}})),}$

what can be justified with the Graßmann identity . The latter formula defines the so-called vectorial Laplace operator.

presentation

In two dimensions

For a function in Cartesian coordinates , the application of the Laplace operator gives ${\ displaystyle f}$ ${\ displaystyle (x, y)}$

${\ displaystyle \ Delta f = {\ frac {\ partial ^ {2} f} {\ partial x ^ {2}}} + {\ frac {\ partial ^ {2} f} {\ partial y ^ {2} }}.}$

In polar coordinates results ${\ displaystyle (r, \ varphi)}$

${\ displaystyle \ Delta f = {\ frac {\ partial ^ {2} f} {\ partial r ^ {2}}} + {\ frac {1} {r}} {\ frac {\ partial f} {\ partial r}} + {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2} f} {\ partial \ varphi ^ {2}}}}$

or

${\ displaystyle \ Delta f = {\ frac {1} {r}} {\ frac {\ partial} {\ partial r}} \ left (r \, {\ frac {\ partial f} {\ partial r}} \ right) + {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2} f} {\ partial \ varphi ^ {2}}}.}$

In three dimensions

For a function with three variables results in Cartesian coordinates${\ displaystyle f}$ ${\ displaystyle (x, y, z)}$

${\ displaystyle \ Delta f = {\ frac {\ partial ^ {2} f} {\ partial x ^ {2}}} + {\ frac {\ partial ^ {2} f} {\ partial y ^ {2} }} + {\ frac {\ partial ^ {2} f} {\ partial z ^ {2}}}.}$

In cylindrical coordinates this results ${\ displaystyle (\ rho, \ varphi, z)}$

${\ displaystyle \ Delta f = {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} \ left (\ rho \, {\ frac {\ partial f} {\ partial \ rho}} \ right) + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial ^ {2} f} {\ partial \ varphi ^ {2}}} + {\ frac {\ partial ^ {2} f} {\ partial z ^ {2}}}}$

and in spherical coordinates ${\ displaystyle (r, \ theta, \ varphi)}$

${\ displaystyle \ Delta f = {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial r}} \ left (r ^ {2} \, {\ frac {\ partial f} {\ partial r}} \ right) + {\ frac {1} {r ^ {2} \ sin \ theta}} {\ frac {\ partial} {\ partial \ theta}} \ left (\ sin \ theta \, {\ frac {\ partial f} {\ partial \ theta}} \ right) + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2} f} {\ partial \ varphi ^ {2}}}.}$

The derivatives of the products in this illustration can still be developed , with the first and second terms changing. The first (radial) term can be written in three equivalent forms:

${\ displaystyle {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial r}} \ left (r ^ {2} \, {\ frac {\ partial f} {\ partial r}} \ right) = {\ frac {\ partial ^ {2} f} {\ partial r ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial f} { \ partial r}} = {\ frac {1} {r}} {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} {\ Big (} rf (r) {\ Big) }}$

These representations of the Laplace operator in cylindrical and spherical coordinates only apply to the scalar Laplace operator. For the Laplace operator, which acts on vector-valued functions, further terms must be taken into account, see the section “ Application to vector fields ” below .

In curvilinear orthogonal coordinates

In any curvilinear orthogonal coordinates , for example in spherical polar coordinates , cylinder coordinates or elliptical coordinates , on the other hand, the more general relationship applies to the Laplace operator

${\ displaystyle \ Delta f = {\ rm {div \, \, grad \, \,}} f = {\ frac {1} {a_ {1} a_ {2} a_ {3}}} \, \, {\ frac {\ partial} {\ partial u_ {1}}} \ left ({\ frac {a_ {2} a_ {3} \, \ partial f} {a_ {1} \, \ partial u_ {1} }} \ right) + {\ frac {1} {a_ {1} a_ {2} a_ {3}}} \, \, {\ frac {\ partial} {\ partial u_ {2}}} \ left ( {\ frac {a_ {1} a_ {3} \, \ partial f} {a_ {2} \, \ partial u_ {2}}} \ right) + {\ frac {1} {a_ {1} a_ { 2} a_ {3}}} \, \, {\ frac {\ partial} {\ partial u_ {3}}} \ left ({\ frac {a_ {1} a_ {2} \, \ partial f} { a_ {3} \, \ partial u_ {3}}} \ right)}$

with the through

${\ displaystyle \ mathrm {d} {\ vec {r}} = \ sum _ {i = 1} ^ {3} \, a_ {i} \, {\ hat {e}} _ {i} (u_ { 1}, u_ {2}, u_ {3}) \, \ mathrm {d} u_ {i}}$
${\ displaystyle {\ rm {grad \, \,}} f = \ sum _ {i = 1} ^ {3} \, {\ frac {\ partial f} {a_ {i} \, \ partial u_ {i }}} \, {\ hat {e}} _ {i}}$
${\ displaystyle {\ hat {e}} _ {i} \ cdot {\ hat {e}} _ {k} = \ delta _ {i, k} = {\ begin {cases} 1 & {\ text {for} } i = k \\ 0 & {\ text {for}} i \ neq k \ end {cases}}}$

implies defined sizes . It is not the , but the quantities that have the physical dimension of a "length", whereby it should be noted that they are not constant, but can depend on , and . ${\ displaystyle a_ {i}, u_ {i}, {\ hat {e}} _ {i}}$${\ displaystyle \ mathrm {d} u_ {i}}$${\ displaystyle \ mathrm {d} l_ {i}: = a_ {i} \ cdot \ mathrm {d} u_ {i}}$${\ displaystyle a_ {i}}$${\ displaystyle u_ {1}}$${\ displaystyle u_ {2}}$${\ displaystyle u_ {3}}$

The Laplace-Beltrami relationship applies to even more general coordinates .

Application to vector fields

In a Cartesian coordinate system with -, - and - coordinates and basis vectors : ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle {\ hat {e}} _ {x, y, z}}$

${\ displaystyle \ Delta {\ vec {v}} = {\ frac {\ partial ^ {2}} {\ partial x ^ {2}}} {\ vec {v}} + {\ frac {\ partial ^ { 2}} {\ partial y ^ {2}}} {\ vec {v}} + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}}} {\ vec {v}} = \ Delta v_ {x} {\ hat {e}} _ {x} + \ Delta v_ {y} {\ hat {e}} _ {y} + \ Delta v_ {z} {\ hat {e}} _ {z}}$

When using cylindrical or spherical coordinates, the differentiation of the basic vectors must be taken into account. It results in cylindrical coordinates${\ displaystyle (\ rho, \ varphi, z)}$

${\ displaystyle \ Delta {\ vec {v}} = \ left (\ Delta v _ {\ rho} - {\ frac {1} {\ rho ^ {2}}} v _ {\ rho} - {\ frac {2 } {\ rho ^ {2}}} {\ frac {\ partial v _ {\ varphi}} {\ partial \ varphi}} \ right) {\ hat {e}} _ {\ rho} + \ left (\ Delta v _ {\ varphi} - {\ frac {1} {\ rho ^ {2}}} v _ {\ varphi} + {\ frac {2} {\ rho ^ {2}}} {\ frac {\ partial v_ { \ rho}} {\ partial \ varphi}} \ right) {\ hat {e}} _ {\ varphi} + \ Delta v_ {z} {\ hat {e}} _ {z}}$

and in spherical coordinates ${\ displaystyle (r, \ varphi, \ theta)}$

{\ displaystyle {\ begin {aligned} \ Delta {\ vec {v}} = & \ left (\ Delta v_ {r} - {\ frac {2} {r ^ {2}}} v_ {r} - { \ frac {2} {r ^ {2} \ sin \ theta}} {\ frac {\ partial v _ {\ varphi}} {\ partial \ varphi}} - {\ frac {2} {r ^ {2}} } {\ frac {\ partial v _ {\ theta}} {\ partial \ theta}} - {\ frac {2} {r ^ {2} \ tan \ theta}} v _ {\ theta} \ right) {\ hat {e}} _ {r} \\ & + \ left (\ Delta v _ {\ theta} + {\ frac {2} {r ^ {2}}} {\ frac {\ partial v_ {r}} {\ partial \ theta}} - {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial v _ {\ varphi}} {\ partial \ varphi} } - {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta} \ right) {\ hat {e}} _ {\ theta} \\ & + \ left (\ Delta v _ {\ varphi} + {\ frac {2} {r ^ {2} \ sin \ theta}} {\ frac {\ partial v_ {r}} {\ partial \ varphi}} - {\ frac { 1} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi} + {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial v _ {\ theta}} {\ partial \ varphi}} \ right) {\ hat {e}} _ {\ varphi} \,. \ end {aligned}}}

The terms added to the Laplace derivatives of the vector components result from the derivatives of the basis vectors.

 proof Be in cylindrical coordinates${\ displaystyle (\ rho, \ varphi, z)}$ ${\ displaystyle {\ hat {e}} _ {\ rho} = {\ begin {pmatrix} \ cos \ varphi \\\ sin \ varphi \\ 0 \ end {pmatrix}}, \ quad {\ hat {e} } _ {\ varphi} = {\ begin {pmatrix} - \ sin \ varphi \\\ cos \ varphi \\ 0 \ end {pmatrix}}, \ quad {\ hat {e}} _ {z} = {\ begin {pmatrix} 0 \\ 0 \\ 1 \ end {pmatrix}}}$ taken as orthonormal basis vectors. Their derivatives are: Here, as in the following, an index after a comma means a derivative according to the specified coordinate, for example the application of the Laplace operator to a vector field results in: i.e. the formula specified in the text. ${\ displaystyle {\ hat {e}} _ {\ rho, \ varphi} = {\ hat {e}} _ {\ varphi} \ quad {\ text {and}} \ quad {\ hat {e}} _ {\ varphi, \ varphi} = - {\ hat {e}} _ {\ rho}}$ ${\ displaystyle {\ hat {e}} _ {\ rho, \ varphi}: = {\ frac {\ partial} {\ partial \ varphi}} {\ hat {e}} _ {\ rho}.}$ ${\ displaystyle \ Delta = {\ frac {\ partial ^ {2}} {\ partial \ rho ^ {2}}} + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}}}}$ {\ displaystyle {\ begin {aligned} & \ left ({\ frac {\ partial ^ {2}} {\ partial \ rho ^ {2}}} + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}}} \ right) (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) \\ = & {\ frac {\ partial ^ {2}} {\ partial \ rho ^ {2 }}} (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e} } _ {z}) + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) \\ & + {\ frac {1} {\ rho ^ { 2}}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}} } (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) \\ = & v _ {\ rho, \ rho \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho \ rho} {\ hat {e}} _ {\ varphi} + v_ {z, \ rho \ rho} {\ hat {e}} _ {z} + {\ frac {1} {\ rho}} (v _ {\ rho, \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho} {\ hat {e}} _ {\ varphi } + v_ {z, \ rho} {\ hat {e}} _ {z}) \\ & + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial} {\ partial \ varphi}} (v _ {\ rho, \ varphi} {\ hat {e}} _ {\ rho} + v _ {\ rho} {\ hat {e}} _ {\ varphi} + v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ varphi} -v _ {\ varphi} {\ hat {e}} _ {\ rho} + v_ {z, \ varphi} {\ hat {e}} _ {z }) \\ & + v _ {\ rho, zz} {\ hat {e}} _ {\ rho} + v _ {\ varphi, zz} {\ hat {e}} _ {\ varphi} + v_ {z, zz} {\ hat {e}} _ {z} \\ = & v _ {\ rho, \ rho \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho \ rho} { \ hat {e}} _ {\ varphi} + v_ {z, \ rho \ rho} {\ hat {e}} _ {z} + {\ frac {1} {\ rho}} (v _ {\ rho, \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho} {\ hat {e}} _ {\ varphi} + v_ {z, \ rho} {\ hat {e} } _ {z}) \\ & + {\ frac {1} {\ rho ^ {2}}} (v _ {\ rho, \ varphi \ varphi} {\ hat {e}} _ {\ rho} + 2v_ {\ rho, \ varphi} {\ hat {e}} _ {\ varphi} -v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ varphi \ varphi} {\ hat {e}} _ {\ varphi} -2v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ rho} -v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z, \ varphi \ varphi} {\ hat {e}} _ {z}) \\ & + v _ {\ rho , zz} {\ hat {e}} _ {\ rho} + v _ {\ varphi, zz} {\ hat {e}} _ {\ varphi} + v_ {z, zz} {\ hat {e}} _ {z} \\ = & + \ left (\ Delta v _ {\ rho} - {\ frac {1} {\ rho ^ {2}}} v _ {\ rho} - {\ frac {2} {\ rho ^ {2}}} v _ {\ varphi, \ varphi} \ right) {\ hat {e}} _ {\ rho} + \ left (\ Delta v _ {\ varphi} - {\ frac {1} {\ rho ^ {2}}} v _ {\ varphi} + {\ frac {2} {\ rho ^ {2}}} v _ {\ rho, \ varphi} \ right) {\ hat {e}} _ {\ varphi} + \ Delta v_ {z} {\ hat {e}} _ {z}, \ end {aligned}}} The basis vectors ${\ displaystyle {\ hat {e}} _ {r} = {\ begin {pmatrix} \ sin \ theta \ cos \ varphi \\\ sin \ theta \ sin \ varphi \\\ cos \ theta \ end {pmatrix} }, \ qquad {\ hat {e}} _ {\ theta} = {\ begin {pmatrix} \ cos \ theta \ cos \ varphi \\\ cos \ theta \ sin \ varphi \\ - \ sin \ theta \ end {pmatrix}}, \ qquad {\ hat {e}} _ {\ varphi} = {\ begin {pmatrix} - \ sin \ varphi \\\ cos \ varphi \\ 0 \ end {pmatrix}}}$ be used. These vectors gave the derivatives Application of the Laplace operator to a vector field: thus the same result as given in the text. {\ displaystyle {\ begin {aligned} {\ hat {e}} _ {r, \ theta} = & {\ begin {pmatrix} \ cos \ theta \ cos \ varphi \\\ cos \ theta \ sin \ varphi \ \ - \ sin \ theta \ end {pmatrix}} = {\ hat {e}} _ {\ theta} \ ,, \ quad {\ hat {e}} _ {r, \ varphi} = {\ begin {pmatrix } - \ sin \ theta \ sin \ varphi \\\ sin \ theta \ cos \ varphi \\ 0 \ end {pmatrix}} = \ sin \ theta {\ hat {e}} _ {\ varphi} \\ {\ hat {e}} _ {\ theta, \ theta} = & {\ begin {pmatrix} - \ sin \ theta \ cos \ varphi \\ - \ sin \ theta \ sin \ varphi \\ - \ cos \ theta \ end {pmatrix}} = - {\ hat {e}} _ {r} \ ,, \ quad {\ hat {e}} _ {\ theta, \ varphi} = {\ begin {pmatrix} - \ cos \ theta \ sin \ varphi \\\ cos \ theta \ cos \ varphi \\ 0 \ end {pmatrix}} = \ cos \ theta {\ hat {e}} _ {\ varphi} \\ {\ hat {e}} _ { \ varphi, \ varphi} = & {\ begin {pmatrix} - \ cos \ varphi \\ - \ sin \ varphi \\ 0 \ end {pmatrix}} = {\ hat {e}} _ {z} \ times { \ hat {e}} _ {\ varphi} = - \ sin \ theta {\ hat {e}} _ {r} - \ cos \ theta {\ hat {e}} _ {\ theta} \ end {aligned} }} ${\ displaystyle \ Delta = {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r} } + {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {r ^ {2 } \ tan \ theta}} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}}}$ {\ displaystyle {\ begin {aligned} & \ left ({\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} + {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac { 1} {r ^ {2} \ tan \ theta}} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta} } {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} \ right) \ cdot (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi}) \\ = & {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat { e}} _ {\ varphi}) + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi}) + {\ frac {1} {r ^ {2}} } {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e }} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac {1} {r ^ {2} \ tan \ theta}} {\ frac {\ partial} {\ partial \ theta}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ v arphi}) + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} ) \\ = & v_ {r, rr} {\ hat {e}} _ {r} + v _ {\ theta, rr} {\ hat {e}} _ {\ theta} + v _ {\ varphi, rr} { \ hat {e}} _ {\ varphi} + {\ frac {2} {r}} v_ {r, r} {\ hat {e}} _ {r} + {\ frac {2} {r}} v _ {\ theta, r} {\ hat {e}} _ {\ theta} + {\ frac {2} {r}} v _ {\ varphi, r} {\ hat {e}} _ {\ varphi} \ \ & + {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial \ theta}} (v_ {r, \ theta} {\ hat {e}} _ {r} + v_ {r} {\ hat {e}} _ {\ theta} + v _ {\ theta, \ theta} {\ hat {e}} _ {\ theta} -v _ {\ theta} {\ hat {e} } _ {r} + v _ {\ varphi, \ theta} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac {1} {r ^ {2} \ tan \ theta}} ( v_ {r, \ theta} {\ hat {e}} _ {r} + v_ {r} {\ hat {e}} _ {\ theta} + v _ {\ theta, \ theta} {\ hat {e} } _ {\ theta} -v _ {\ theta} {\ hat {e}} _ {r} + v _ {\ varphi, \ theta} {\ hat {e}} _ {\ varphi}) \\ & + { \ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial} {\ partial \ varphi}} (v_ {r, \ varphi} {\ hat {e}} _ {r} + \ sin \ theta v_ {r} {\ hat {e}} _ {\ varphi} + v _ {\ theta, \ varphi} {\ hat {e }} _ {\ theta} + \ cos \ theta v _ {\ theta} {\ hat {e}} _ {\ varphi} + v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ varphi} - \ sin \ theta v _ {\ varphi} {\ hat {e}} _ {r} - \ cos \ theta v _ {\ varphi} {\ hat {e}} _ {\ theta}) \\ = & v_ {r , rr} {\ hat {e}} _ {r} + v _ {\ theta, rr} {\ hat {e}} _ {\ theta} + v _ {\ varphi, rr} {\ hat {e}} _ {\ varphi} + {\ frac {2} {r}} v_ {r, r} {\ hat {e}} _ {r} + {\ frac {2} {r}} v _ {\ theta, r} {\ hat {e}} _ {\ theta} + {\ frac {2} {r}} v _ {\ varphi, r} {\ hat {e}} _ {\ varphi} \\ & + {\ frac { 1} {r ^ {2}}} (v_ {r, \ theta \ theta} {\ hat {e}} _ {r} + v_ {r, \ theta} {\ hat {e}} _ {\ theta } + v_ {r, \ theta} {\ hat {e}} _ {\ theta} -v_ {r} {\ hat {e}} _ {r} + v _ {\ theta, \ theta \ theta} {\ hat {e}} _ {\ theta} -v _ {\ theta, \ theta} {\ hat {e}} _ {r} -v _ {\ theta, \ theta} {\ hat {e}} _ {r} -v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi, \ theta \ theta} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac { 1} {r ^ {2} \ tan \ theta}} (v_ {r, \ theta} {\ hat {e}} _ {r} + v_ {r} {\ hat {e}} _ {\ theta} + v _ {\ theta, \ theta} {\ hat {e}} _ {\ theta} -v _ {\ theta} {\ hat {e}} _ {r} + v _ {\ varphi, \ theta} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} (v_ {r, \ varphi \ varphi} {\ hat {e}} _ {r} + \ sin \ theta v_ {r, \ varphi} {\ hat {e}} _ {\ varphi} + \ sin \ theta v_ {r, \ varphi} {\ hat {e}} _ {\ varphi} - \ sin ^ {2} \ theta v_ {r} {\ hat {e}} _ {r} - \ sin \ theta \ cos \ theta v_ {r } {\ hat {e}} _ {\ theta} \\ & + v _ {\ theta, \ varphi \ varphi} {\ hat {e}} _ {\ theta} + \ cos \ theta v _ {\ theta, \ varphi} {\ hat {e}} _ {\ varphi} + \ cos \ theta v _ {\ theta, \ varphi} {\ hat {e}} _ {\ varphi} - \ sin \ theta \ cos \ theta v_ { \ theta} {\ hat {e}} _ {r} - \ cos ^ {2} \ theta v _ {\ theta} {\ hat {e}} _ {\ theta} \\ & + v _ {\ varphi, \ varphi \ varphi} {\ hat {e}} _ {\ varphi} - \ sin \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {r} - \ cos \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ theta} - \ sin \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {r} - \ sin ^ {2} \ theta v_ { \ varphi} {\ hat {e}} _ {\ varphi} \\ & - \ cos \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ theta} - \ cos ^ {2} \ theta v _ {\ varphi} {\ hat {e}} _ {\ varphi}) \\ = & {\ Bigl (} v_ {r, rr} + {\ frac {2} {r}} v_ {r, r} + {\ frac {1} {r ^ {2}}} v_ {r, \ theta \ theta} + {\ frac {1} {r ^ {2} \ tan \ theta}} v_ {r, \ theta} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v_ {r, \ varphi \ varphi} \\ & \ qquad - {\ frac {1} {r ^ {2}}} v_ {r} - {\ frac {1} {r ^ {2}}} v _ {\ theta, \ theta } - {\ frac {1} {r ^ {2}}} v _ {\ theta, \ theta} - {\ frac {1} {r ^ {2} \ tan \ theta}} v _ {\ theta} - { \ frac {1} {r ^ {2}}} v_ {r} - {\ frac {\ cos \ theta} {r ^ {2} \ sin \ theta}} v _ {\ theta} - {\ frac {1 } {r ^ {2} \ sin \ theta}} v _ {\ varphi, \ varphi} - {\ frac {1} {r ^ {2} \ sin \ theta}} v _ {\ varphi, \ varphi} {\ Bigr)} {\ hat {e}} _ {r} \\ & + {\ Bigl (} v _ {\ theta, rr} + {\ frac {2} {r}} v _ {\ theta, r} + { \ frac {1} {r ^ {2}}} v _ {\ theta, \ theta \ theta} + {\ frac {1} {r ^ {2} \ tan \ theta}} v _ {\ theta, \ theta} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta, \ varphi \ varphi} \\ & \ qquad + {\ frac {2} {r ^ {2 }}} v_ {r, \ theta} - {\ frac {1} {r ^ {2}}} v _ {\ theta} + {\ frac {1} {r ^ {2} \ tan \ theta}} v_ {r} - {\ frac {\ cos \ theta} {r ^ {2} \ sin \ theta}} v_ {r} - {\ frac {\ cos ^ {2} \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta} - {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi, \ varphi} { \ Bigr)} {\ hat {e}} _ {\ theta} \\ & + {\ Bigl (} v _ {\ varphi, rr} + {\ frac {2} {r}} v _ {\ varphi, r} + {\ frac {1} {r ^ {2}}} v _ {\ varphi, \ theta \ theta } + {\ frac {1} {r ^ {2} \ tan \ theta}} v _ {\ varphi, \ theta} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta} } v _ {\ varphi, \ varphi \ varphi} \\ & \ qquad + {\ frac {2} {r ^ {2} \ sin \ theta}} v_ {r, \ varphi} + {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta, \ varphi} - {\ frac {\ sin ^ {2} \ theta + \ cos ^ {2} \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi} {\ Bigr)} {\ hat {e}} _ {\ varphi} \\ = & \ left (\ Delta v_ {r } - {\ frac {2} {r ^ {2}}} v_ {r} - {\ frac {2} {r ^ {2}}} v _ {\ theta, \ theta} - {\ frac {2} {r ^ {2} \ tan \ theta}} v _ {\ theta} - {\ frac {2} {r ^ {2} \ sin \ theta}} v _ {\ varphi, \ varphi} \ right) {\ hat {e}} _ {r} \\ & + \ left (\ Delta v _ {\ theta} + {\ frac {2} {r ^ {2}}} v_ {r, \ theta} - {\ frac {1 } {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta} - {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v_ {\ varphi, \ varphi} \ right) {\ hat {e}} _ {\ theta} \\ & + \ left (\ Delta v _ {\ varphi} + {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta, \ varphi} - {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi} + {\ frac {2} {r ^ {2} \ sin \ theta}} v_ {r, \ varphi} \ right) {\ hat {e}} _ {\ varphi} \ end {aligned}},}

properties

The Laplace operator is a linear operator , that is: If and are twice differentiable functions and and constants, then we have ${\ displaystyle f}$${\ displaystyle g}$${\ displaystyle a}$${\ displaystyle b}$

${\ displaystyle \ Delta (a \ cdot f + b \ cdot g) = a \ cdot (\ Delta f) + b \ cdot (\ Delta g).}$

As for other linear differential operators, a generalized product rule applies to the Laplace operator . This reads

${\ displaystyle \ Delta (fg) = f \ Delta g + 2 \ langle \ nabla f, \ nabla g \ rangle + g \ Delta f,}$

where there are two twice continuously differentiable functions with and is the standard Euclidean scalar product. ${\ displaystyle f, g \ colon U \ to \ mathbb {R}}$${\ displaystyle U \ subset \ mathbb {R} ^ {n}}$${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$

The Laplace operator is rotationally symmetric , that is: If there is a twice differentiable function and a rotation , then applies ${\ displaystyle f}$${\ displaystyle R}$

${\ displaystyle \ left (\ Delta f \ right) \ circ R = \ Delta \ left (f \ circ R \ right),}$

where " " stands for the concatenation of images. ${\ displaystyle \ circ}$

The main symbol of the Laplace operator is . So it is a second order elliptic differential operator . It follows from this that it is a Fredholm operator and, by means of Atkinson's theorem, it follows that it is right and left invertible modulo of a compact operator . ${\ displaystyle - \ | \ xi \ | ^ {2}}$

The Laplace operator

${\ displaystyle - \ Delta \ colon {\ mathcal {S}} (\ mathbb {R} ^ {n}) \ rightarrow L ^ {2} (\ mathbb {R} ^ {n})}$

on the Schwartz space is essentially self-adjoint . He therefore has a degree

${\ displaystyle - \ Delta \ colon H ^ {2} (\ mathbb {R} ^ {n}) \ rightarrow L ^ {2} (\ mathbb {R} ^ {n})}$

to a self-adjoint operator on Sobolev space . This operator is also nonnegative, so its spectrum is on the nonnegative real axis, that is: ${\ displaystyle H ^ {2} (\ mathbb {R} ^ {n}) \ subset L ^ {2} (\ mathbb {R} ^ {n})}$

${\ displaystyle \ sigma (- \ Delta) \ subset \ mathbb {R} _ {0} ^ {+}}$
${\ displaystyle - \ Delta f = \ lambda f}$

of the Laplace operator is called the Helmholtz equation . If a restricted area and the Sobolev space with the boundary values ​​is in , then the eigenfunctions of the Laplace operator form a complete orthonormal system of and its spectrum consists of a purely discrete , real point spectrum that can only have one accumulation point . This follows from the spectral theorem for self-adjoint elliptic differential operators. ${\ displaystyle \ Omega \ subset \ mathbb {R} ^ {n}}$${\ displaystyle H_ {0} ^ {2} (\ Omega)}$${\ displaystyle f = 0}$${\ displaystyle \ partial \ Omega}$${\ displaystyle - \ Delta \ colon H_ {0} ^ {2} (\ Omega) \ rightarrow L ^ {2} (\ Omega)}$${\ displaystyle L ^ {2} (\ Omega)}$${\ displaystyle \ infty}$

For a function at a point it clearly shows how the mean value of over concentric spherical shells changes with increasing spherical radius . ${\ displaystyle \ Delta f (p)}$${\ displaystyle f}$${\ displaystyle p}$${\ displaystyle f}$${\ displaystyle p}$${\ displaystyle f (p)}$

Poisson and Laplace equation

definition

The Laplace operator appears in a number of important differential equations. The homogeneous differential equation

${\ displaystyle \ Delta \ varphi = 0}$

is called the Laplace equation and twice continuously differentiable solutions of this equation are called harmonic functions . The corresponding inhomogeneous equation

${\ displaystyle \ Delta \ varphi = f}$

is called Poisson's equation.

Fundamental solution

The fundamental solution of the Laplace operator satisfies the Poisson equation${\ displaystyle G ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime})}$

${\ displaystyle \ Delta \, G ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime}) = \ delta ({\ vec {x}} - {\ vec {x} } ^ {\, \ prime})}$

with the delta distribution on the right. This function depends on the number of room dimensions. ${\ displaystyle \ delta}$

${\ displaystyle G ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime}) = - {\ frac {1} {4 \ pi \ | {\ vec {x}} - {\ vec {x}} ^ {\, \ prime} \ |}} + F ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime})}$ With ${\ displaystyle \ Delta \, F ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime}) = 0}$

This fundamental solution is required in electrodynamics as an aid to solving boundary value problems .

${\ displaystyle G ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime}) = {\ frac {\ ln (\ | {\ vec {x}} - {\ vec { x}} ^ {\, \ prime} \ |)} {2 \ pi}} + F ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime})}$ With ${\ displaystyle \ Delta \, F ({\ vec {x}}, {\ vec {x}} ^ {\, \ prime}) = 0}$

Generalizations

D'Alembert operator

The Laplace operator, together with the second time derivative, gives the D'Alembert operator:

${\ displaystyle \ square = {\ frac {1} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} - \ Delta}$

This operator can be viewed as a generalization of the Laplace operator to Minkowski space . ${\ displaystyle \ Delta}$

Generalized Laplace operator

For the Laplace operator, which was originally always understood as the operator of Euclidean space, with the formulation of Riemannian geometry there was the possibility of generalization to curved surfaces and Riemannian or pseudo-Riemannian manifolds . This more general operator is known as the generalized Laplace operator.

Discrete Laplace operator

The Laplace operator is applied to a discrete input function g n or g nm via a convolution . You can use the following simple convolution masks:

1D filter ${\ displaystyle \ quad {\ vec {D}} _ {x} ^ {2} \; = {\ begin {bmatrix} 1 & -2 & 1 \ end {bmatrix}}}$
2D filter: ${\ displaystyle \ quad \ mathbf {D} _ {xy} ^ {2} = {\ begin {bmatrix} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \ end {bmatrix}}}$

There are alternative variants for two dimensions that also take diagonal edges into account, for example:

2D filter: ${\ displaystyle \ quad \ mathbf {D} _ {xy} ^ {2} = {\ begin {bmatrix} 1 & 1 & 1 \\ 1 & -8 & 1 \\ 1 & 1 & 1 \ end {bmatrix}}}$

These convolution masks are obtained by discretizing the difference quotients. The Laplace operator corresponds to a weighted sum over the value at neighboring points. The edge detection in the image processing (see Laplacian filter ) is a possible field of application of discrete Laplace operators. An edge appears there as the zero crossing of the second derivative of the signal. Discrete Laplace operators are also used in the discretization of differential equations or in graph theory.

Applications

literature

• Bronstein, Semendjajew, Musiol, Mühlig: Taschenbuch der Mathematik. Harri Deutsch, 1999, 4th edition, ISBN 3-8171-2004-4 .
• Otto Forster : Analysis. Volume 3: Measure and integration theory, integral theorems in R n and applications , 8th improved edition. Springer Spectrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5 .
• Russell Merris: Laplacian matrices of graphs: a survey. In: Linear Algebra and its Applications. 197-198, 143-176 (1994). ISSN 0024-3795