Second Derivatives - & Live Lecture 1

SECOND DERIVATIVES we defined gradient, divergence and curl; the 3 possible first derivatives Now Consider gradient of some scalar field ∇T (which is a vector) we can have div (∇T) and curl (∇T) →Now curl of a gradient is always zero (MA III) *→ ucan also take the gradient of divergence div (curd F), and curl (curl F) always zero (for F defined in simple connected) (1) divergance of Gradient ∇⋅(∇T)=∂²T/∂x²+∂²T/∂y²+∂²T/∂z²=∇²T Laplacian Laplacin of a scalar field ∇²T=∂²T/∂x²+∂²T/∂y²+∂²T/∂z² Laplauan of a vector field ∇²v=x̂(∇²v)+ŷ(∇²v)+ẑ(∇²v) Laplauan of avector field is a vector Curl of the Gradient → Gradient fields are always cons crvative hence cud of Gradient field of any scalar vector is zero. (provided the scalar field is continuous) (ii) gradient of divergence (Not to be confused with →for a vector field ∇(∇⋅A) laplacian of avector) =(∂²ax/∂x²+∂²ay/∂y∂x+∂²az/∂x∂z)x̂+(∂²ay/∂x∂y+∂²ay/∂y²+∂²az/∂z∂y)ŷ +(∂²az/∂z∂x+∂²az/∂z∂y+∂²az/∂ (iv) divergence of a curl Now we have: ∇⋅(∇×A) (∂/∂x, ∂/∂y, ∂/∂z) * ((∂az/∂y - ∂ay/∂z)i + (∂ax/∂z - ∂az/∂x)j + (∂ay/∂x - ∂ax/∂y)k) = ∂az/∂xy - ∂ay/∂zx + ∂ax/∂zy - ∂az/∂yx + ∂ay/∂zx - ∂ax/∂yz = 0 always (thus this product is meaningless) (v) curl of the curl of a vector field We have: ∇x(∇xA) Now let A = axi + ayj + azk ∂/∂y ∂/∂z ∂/∂x ∂/∂y ∂/∂z (∂az/∂y - ∂ay/∂z) (∂ax/∂z - ∂az/∂x) (∂ay/∂x - ∂ax/∂y) = i [∂²ay/∂yx - ∂²ax/∂y² - ∂²az/∂z² + ∂²az/∂zy] + j [∂²az/∂yx - ∂²ax/∂x² - ∂²ay/∂z² + ∂²ay/∂zx] + k [∂²ax/∂zx - ∂²az/∂x² - ∂²ay/∂y² + ∂²ay/∂zy] Live Lecture 1 (week 1) vector analysis 24/3/2022 → Geometrical interpretation of gradient of scalar field direction: is along the maximum variation of T at any given point ie its perpendicular to the contour plot at that point → dT along. Magnitude the above direction. dl scalar field f(xy) (f) (direction of f at point P) → Variation of vector in space (field) is in 2 ways" divergence and rotation/curl vector plot Positive no div variation along vecbr direction (source) negative div (sink) →nodivergance (this is also divergance) F(xiy) contourplot