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Vector Calculus (Chapter 13)

Vector Calculus (Chapter 13). Vector Calculus. Chapter. 13. 13.2-13.3. 13.4. 13.1. Scalar Fields, 2D Vector Fields. Gradient Vector Fields. Line Integrals. Green’s Theorem. F (x,y)= < P(x,y),Q(x,y) >. Scalar Fields and Vector Fields.

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Vector Calculus (Chapter 13)

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  1. Vector Calculus(Chapter 13)

  2. Vector Calculus Chapter 13 13.2-13.3 13.4 13.1 ScalarFields, 2D VectorFields Gradient VectorFields LineIntegrals Green’sTheorem

  3. F(x,y)=<P(x,y),Q(x,y)>

  4. Scalar Fields and Vector Fields • The simplest possible physical field is the scalar field. • It represents a function depending on the position in space. A scalar field is characterized at each point in space by a single number. • Examples of scalar fields • temperature, gravitational potential, electrostatic potential (voltage)

  5. Scalar Fields Visualization of z=V(x,y) • Scalar potential function for a dipole V(x,y)

  6. Maple commands

  7. Scalar Fields and Equipotential Lines • The level curves or contours of the function z=V(x,y) are the equipotential lines of the scalar potential field V(x,y)

  8. The Gradient defines a Vector Field (the force field)

  9. Arrow Diagram for Vector Field

  10. Direction Field (magnitude=1)

  11. Equipotential surfaces are orthogonal to the electric force field Notice the force field is directed towards placeswhere the potential V is lower, e.g., where thecharge is negative - at(0.25,0).But mathematically,the gradient points in the opposite direction (greatest ascent) which is why f=-Vand F=grad(f)=grad(-V)

  12. 2D vector field visualization of the flow field past an air foil using arrows

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