3. Differential operators Gradient 1- Definition. f(x,y,z) is a differentiable scalar field 2 – Physical meaning: is the local variation of f along dr. Particularly, grad f is perpendicular to the line f = ctt. Summer School 2007 B. Rossetto

3. Differential operators Divergence 1 – Definition is a differentiable vector field x x+dx 2 – Physical meaning is associated to local conservation laws: for example, we’ll show that if the mass of fluid (or of charge) outcoming from a domain is equal to the mass entering, then is the fluid velocity (or the current) vectorfield Summer School 2007 B. Rossetto

3. Differential operators Curl 1 – Definition. is a differentiable vector field 2 – Physical meaning: is related to the local rotation of the vectorfield: If is the fluid velocity vectorfield Summer School 2007 B. Rossetto

3. Differential operators Laplacian: definitions 1 – Scalar Laplacian. f(x,y,z) is a differentiable scalar field 2 – Vector Laplacian. is a differentiable vector field Summer School 2007 B. Rossetto

3. Differential operators Laplacian: physical meaning f(x) As a second derivative, the one-dimensional Laplacian operator is related to minima and maxima: when the second derivative is positive (negative), the curvature is concave (convexe). convex concave x In most of situations, the 2-dimensional Laplacian operator is also related to local minima and maxima. If vE is positive: E Summer School 2007 B. Rossetto

3. Differential operators Summary Operator grad div curl Laplacian is a vector a scalar (resp. a vector) concerns a scalar field a vector field (resp. a vector field) Definition resp. Summer School 2007 B. Rossetto

3. Differential operators Cylindrical coordinates Summer School 2007 B. Rossetto

3. Differential operators Cylindrical coordinates Summer School 2007 B. Rossetto

3. Differential operators Spherical coordinates Summer School 2007 B. Rossetto

3. Differential operators Conservative vectorfield Theorem. If there exists f such that then H P Consequently, the value of the integral doesn’t depend on the path, but only on its beginning A and its end B. We say that the vectorfield is conservative Proof. Summer School 2007 B. Rossetto

3. Differential operators 1st Stokes formula: vectorfield global circulation Theorem. If S(C) is any oriented surface delimited by C: S(C) C Sketch of proof. y Vy . Vx . . P x . … and then extend to any surface delimited by C. Summer School 2007 B. Rossetto

3. Differential operators 2nd Stokes formula: global conservation laws Theorem. If V(C) is the volume delimited by S Sketch of proof. Flow through the oriented elementary planes x = ctt and x+dx = ctt : x x+dx -Vx(x,y,z).dydz + Vx (x+dx,y,z).dydz and then extend this expression to the lateral surface of the cube. Other expression: extended to the vol. of the elementary cube: Summer School 2007 B. Rossetto

3. Differential operators Vector identities Use Einstein convention and Levi-Civita symbol to show them curl(grad f) = 0 Summer School 2007 B. Rossetto