EE8391 EMT Unit I Vector Calculus
- 1. Vector Calculus K. Karthikeyan, Associate Professor, Department of Electrical and Electronics Engineering, Ramco Institute of Technology
- 2. LINE INTEGRAL If the path of integration is a closed curve such as abca in Figure, eq. becomes a closed contour integral 09-03-2022 2
- 3. SURFACE INTEGRAL Given a vector field A, continuous in a region containing the smooth surface S, we define the surface integral or the flux of A through S (see Figure) as 09-03-2022 3 At any point on S, an is the unit normal to S
- 4. SURFACE INTEGRAL 09-03-2022 4 For a closed surface (defining a volume), eq. becomes which is referred to as the net outward flux of A from S. Notice that a closed path defines an open surface whereas a closed surface defines a volume.
- 5. VOLUME INTEGRAL 09-03-2022 5 Volume integral of the scalar ρv over the volume v.
- 6. DEL OPERATOR • The del operator, written ∇ , is the vector differential operator. • In Cartesian coordinates, • This vector differential operator, otherwise known as the gradient operator, is not a vector in itself, • but when it operates on a scalar function, for example, a vector ensues. 09-03-2022 6
- 7. DEL OPERATOR 09-03-2022 7 The ∇ operator is useful in defining,
- 8. GRADIENT OF A SCALAR 09-03-2022 8 The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V.
- 9. DIVERGENCE OF A VECTOR 09-03-2022 9 Physically, we may regard the divergence of the vector field A at a given point as a measure of how much the field diverges or emanates from that point. The divergence of A at a given point P is the outward flux per unit volume as the volume shrinks about P.
- 10. DIVERGENCE OF A VECTOR 09-03-2022 10 The divergence of a vector field can also be viewed as simply the limit of the field's source density; it is positive at a source point in the field, and negative at a sink point, or zero where there is neither sink nor source.
- 11. DIVERGENCE THEOREM 09-03-2022 11 The divergence theorem states that the total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A.
- 12. CURL OF A VECTOR The curl of A is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum. 09-03-2022 12
- 13. CURL OF A VECTOR 09-03-2022 13
- 14. STOKES THEOREM 09-03-2022 14 Stokes's theorem states that the circulation of a vector field A around a (closed) path L is equal to the surface integral of the curl of A over the open surface S bounded by L, provided A and ∇ × 𝐴 are continuous on S.
- 15. LAPLACIAN OF A SCALAR The Laplacian of a scalar field V, written as ∇2𝑉, is the divergence of the gradient of V. 09-03-2022 15