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3. Differential operators

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. 3. Differential operators. Divergence. 1 – Definition.

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3. Differential operators

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  1. 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, gradf is perpendicular to the line f = ctt. B. Rossetto

  2. 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 B. Rossetto

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

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

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

  6. 3. Differential operators • Summary resp. B. Rossetto

  7. 3. Differential operators • Cylindrical coordinates B. Rossetto

  8. 3. Differential operators • Cylindrical coordinates B. Rossetto

  9. 3. Differential operators • Spherical coordinates B. Rossetto

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

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

  12. 3. Differential operators • 2nd Stokes formula: global conservation laws Theorem. If V(C)is the volumedelimited 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: B. Rossetto

  13. 3. Differential operators • Vector identities Use Einstein convention and Levi-Civita symbol to show them curl(gradf) = 0 B. Rossetto

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