m2415formulas - formulas and calculus theorems

Q: When is a vector field conservative in R^3?

If a vector field F is defined on all of R^3, its component functions have continuous partial derivatives, and curl F = 0, then F is conservative. In that case, div(curl F)=0 is also stated.

Q: Which theorems relate line, surface, and volume integrals in the document?

The document states the Fundamental Theorem of Line Integrals, Green’s Theorem (circulation form) for planar regions, Stokes’ Theorem for oriented surfaces with boundary curves, and the Divergence Theorem connecting flux through a closed surface to a triple integral over the enclosed solid.

Nov 1, 2026

Uploaded by RizkyPratama_92

A collection of calculus formulas and vector calculus results centered on smooth curves, derivatives, and multivariable functions. It includes conditions for curvature-related expressions, the Second Derivative Test for classifying critical points, and multiple integral formulations. The document further presents foundational theorems for line and surface integrals—such as the Fundamental Theorem of Line Integrals, conservative vector field criteria, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem—along with related flux and divergence properties.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages3

Reading Time8 min

AccessFree to download

Derivative-based formulas

Second Derivative Test

Multiple integrals and higher-dimensional expressions

Triple and surface integral setups

Line and surface integral theorems

Fundamental Theorem of Line Integrals

Conservative vector fields

Green’s Theorem (circulation form)

Stokes’ Theorem

Divergence Theorem

􀀌 􀀳®􀀩 􀀌 ■ If 􀀘 is a smooth curve in the plane/space given by 􀁁®(􀁂), where 􀁂 is the arc-length parameter. 􀁞 at 􀁂 is 􀁞 = 􀀌 􀀳􀁂 􀀌 = |®􀀩′(􀁂) ||®􀀩′(􀁃) | |􀁁®′(􀁃) × 􀁁®′′(􀁃) | ■ If 􀀘 is a smooth curve given by 􀁁®(􀁃), then 􀁞 = ~~ ~~ ■ If 􀀘 is a 3-dimensional curve, then 􀁞 = ~~ ~~|􀁁®′(􀁃) | |􀁁®′(􀁃) |3 ■ If 􀀘 is the graph of a function 􀁈 = 􀀵 (􀁇) and both 􀁈 ′ and 􀁈 ′′ exist, then 􀁞 at point (􀁇, 􀁈) is 􀁞 = [1~~ ~~+~~ ~~(|􀁈􀁈′′′)|2~~ ~~]3/2 ■ Second Derivative Test: Let 􀁉 = 􀀵 (􀁇, 􀁈) be a function of two variables for which the first- and second-order partial derivatives are continuous on some disk containing the point (􀁇0 , 􀁈 0 ) . Suppose 􀀵 􀁇 (􀁇0 , 􀁈 0 ) = 0 and 􀀵 􀁈 (􀁇0 , 􀁈 0 ) = 0 (i.e.(􀁇0 , 􀁈 0 ) is a critical point of 􀀵 .) Let 􀀙 = 􀀙 (􀁇0 , 􀁈 0 ) = 􀀵 􀁇 􀁇 (􀁇0 , 􀁈 0 ) 􀀵 􀁈 􀁈 (􀁇0 , 􀁈 0 ) − [ 􀀵 􀁇 􀁈 (􀁇0 , 􀁈 0 )] 2 . i. If 􀀙 > 0 and 􀀵 􀁇 􀁇 (􀁇0 , 􀁈 0 ) > 0, then 􀀵 has a local minimum at (􀁇0 , 􀁈 0 ) . ii. If 􀀙 > 0 and 􀀵 􀁇 􀁇 (􀁇0 , 􀁈 0 ) < 0, then 􀀵 has a local maximum at (􀁇0 , 􀁈 0 ) . iii. If 􀀙 < 0, then 􀀵 has a saddle point at (􀁇0 , 􀁈 0 ) . iv. If 􀀙 = 0, the test is inconclusive. 􀀳􀁈 ~~ 􀁭~~􀁭~~􀀛~~􀁇~~ ~~ 􀀛􀁇 􀁭􀁉 ~~ 􀁭~~􀁭~~􀀛~~􀁇~~ ~~ 􀁭􀁉 ~~ 􀁭~~􀁭~~􀀛~~􀁈 ~~ ~~ = − ~~ ~~ = − ~~ ~~ ~~ ~~ = − ~~ ~~ ~~ ~~ = − ~~ ~~ 􀀳􀁇 ~~ 􀁭~~􀁭~~􀀛~~􀁈~~ ~~ 􀀛􀁈 􀁭􀁇 ~~ 􀁭~~􀁭~~􀀛~~􀁉~~ ~~ 􀁭􀁈 ~~ 􀁭~~􀁭~~􀀛~~􀁉 􀀢􀁇 = ∫ ∫􀀙 􀁈􀁤 (􀁇, 􀁈)􀀳􀀖 􀀢􀁈 = ∫ ∫􀀙 􀁇 􀁤 (􀁇, 􀁈)􀀳􀀖 􀀼 = ∫ ∫􀀙 􀁤 (􀁇, 􀁈)􀀳􀀖􀁇¯ = 􀀢􀀼􀁈 = 1􀀼 ∫ ∫􀀙 􀁇 􀁤 (􀁇, 􀁈)􀀳􀀖 ¯􀁈 = 􀀢􀀼􀁇 = 1􀀼 ∫ ∫􀀙 􀁈􀁤 (􀁇, 􀁈)􀀳􀀖 􀀞􀁇 = ∫ ∫􀀙 􀁈 2􀁤 (􀁇, 􀁈)􀀳􀀖 􀀞􀁈 = ∫ ∫􀀙 􀁇 2􀁤 (􀁇, 􀁈)􀀳􀀖 􀀞0 = ∫ ∫􀀙 (􀁇2 + 􀁈2 )􀁤 (􀁇, 􀁈)􀀳􀀖 􀀢􀁈􀁉 = ∫ ∫ ∫􀀙 􀁇 􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫 􀀢􀁇􀁉 = ∫ ∫ ∫􀀙 􀁈􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫 􀀢􀁇 􀁈 = ∫ ∫ ∫􀀙 􀁉􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫􀁇¯ = 􀀢􀁈􀁉􀀼~~ ~~ ¯􀁈 = 􀀢􀁇􀁉􀀼~~ ~~ ¯􀁉 = 􀀢􀁇~~ ~~􀁈􀀼 􀀞􀁇 = ∫ ∫ ∫􀀚 (􀁈2 + 􀁉 2 )􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫 􀀞􀁈 = ∫ ∫ ∫􀀚 (􀁇2 + 􀁉 2 )􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫 􀀞􀁉 = ∫ ∫ ∫􀀚 (􀁇2 + 􀁈2 )􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫 􀀼 = ∫ ∫ ∫􀀚 􀁤 (􀁇, 􀁈, 􀁉)􀀳􀀫 􀀦 = ∫ ∫ ∫􀀚 􀁦 (􀁇, 􀁈, 􀁉)􀀳􀀫 ∫ ∫􀀧 􀀵 (􀁇, 􀁈)􀀳􀀖 = ∫ ∫􀀨 􀀵 (􀁇 ( 􀁄, 􀁅 ) , 􀁈 ( 􀁄, 􀁅 )) 􀀌 􀁭􀁭~~ ~~((􀁇􀁄~~ ~~,,~~ ~~􀁈􀁅)) 􀀌 􀀳􀁄􀀳􀁅∫ ∫ ∫􀀧 􀀵 (􀁇, 􀁈, 􀁉)􀀳􀀫 = ∫ ∫ ∫􀀨 􀀵 (􀁇 ( 􀁄, 􀁅, 􀁆 ) , 􀁈 ( 􀁄, 􀁅, 􀁆 ) , 􀁉 ( 􀁄, 􀁅, 􀁆 )) 􀀌 􀁭􀁭((􀁇􀁄,,~~ ~~􀁈􀁅,,􀁉􀁆)) 􀀌 􀀳􀁄􀀳􀁅􀀳􀁆 ∫ ∫􀀨 ®􀀛 · 􀀳®􀀨 = ∫ ∫􀀙 ®􀀛 · (􀁁®􀁄 × 􀁁®􀁅)􀀳􀀖∫􀀲 􀀳 ∫􀁕􀁖 ∫􀀰 􀀱 􀀵 ( 􀁤 sin 􀁱 cos 􀁜, 􀁤 sin 􀁱 sin 􀁜, 􀁤 cos 􀁱)􀁤 2 sin 􀁱􀀳􀁤􀀳􀁜􀀳􀁱 􀀖 = ∮􀀘 􀁇􀀳􀁈 = − ∮􀀘 􀁈􀀳􀁇 = 12 ∮􀀘 (􀁇􀀳􀁈 − 􀁈􀀳􀁇) ∫ ∫􀀨 􀀵 (􀁇, 􀁈, 􀁉)􀀳􀀨 = ∫ ∫􀀙 􀀵 (􀁇, 􀁈, 􀀶 (􀁇, 􀁈))√︄ 􀀒 􀁭􀁭􀁉􀁇 􀀓 2 + 􀀒 􀁭􀁭􀁉􀁈 􀀓 2 + 1􀀳􀀖 ∫ ∫􀀨 􀀵 (􀁇, 􀁈, 􀁉)􀀳􀀨 = ∫ ∫􀀙 􀀵 (􀁁®( 􀁄, 􀁅 )) |􀁁®􀁄 × 􀁁®􀁅 |􀀳􀀖 ∫ ∫􀀨 ®􀀛 · 􀀳®􀀨 = ∫ ∫􀀙 􀀒−􀀥 􀁭􀁭􀀶􀁇 − 􀀦 􀁭􀁭􀀶􀁈 + 􀀧􀀓 􀀳􀀖 ∫􀀘 ®􀀛 · 􀀳􀁁® = ∫􀀰 􀀱 ®􀀛(􀁁®(􀁃)) · 􀁁®′(􀁃)􀀳􀁃 = ∫􀀘 ®􀀛 · ®􀀩􀀳􀁂 If ®􀀛 is conservative, then curl®􀀛 = ®0 . 􀀖 (􀀨) = ∫ ∫􀀙 √︄1 + 􀀒 􀁭􀁭􀁉􀁇 􀀓 2 + 􀀒 􀁭􀁭􀁉􀁈 􀀓 2 􀀳􀀖 . ■ The Fundamental Theorem of Line Integrals: Let 􀀘 be any smooth curve given by the vector function 􀁁®(􀁃) . Let 􀀵 be any differentiable function of two or three variables whose gradient vector 􀀵 is continuous on 􀀘 . Then ∫􀀘 􀀵 · 􀀳􀁁® = 􀀵 (􀁁®(􀀱)) − 􀀵 (􀁁®(􀀰)) . ■ If ®􀀛 is a vector field defined on all of R3 whose component functions have continuous partial derivatives and curl ®􀀛 = ®0, then ®􀀛 is a conservative vector field. Also, div(curl®􀀛) = 0. ■ If ®􀀛 is a continuous vector field defined on an oriented surface 􀀨 with unit normal vector ®􀀣, then the surface integral of ®􀀛 over 􀀨 is ∫ ∫ ®􀀛 · 􀀳®􀀨 = ∫ ∫ ®􀀛 · ®􀀣􀀳􀀨 . Also called the flux of ®􀀛 across 􀀨 . ■ Green’s Theorem (Circulation Form): Let 􀀙 be an open, simply connected region with a boundary curve 􀀘 that is apiecewise smooth, simple closed curve oriented counterclockwise. Let ®􀀛 =< 􀀥, 􀀦 > be a vector field with component functions that have continuous partial derivatives on 􀀙 . Then, ∫􀀘 ®􀀛 · 􀀳􀁁® = ∫􀀘 􀀥􀀳􀁇 + 􀀦􀀳􀁈 = ∫ ∫􀀙 (􀀦 􀁇 − 􀀥􀁈 ) 􀀳􀀖 = ∫ ∫􀀙 􀁭􀁭􀀦􀁇 − 􀁭􀁭􀀥􀁈 􀀳􀀖 For flux: ∫􀀘 ®􀀛 · ®􀀣􀀳􀁂 = ∫ ∫􀀙 􀀥 􀁇 + 􀀦􀁈 􀀳􀀖 ■ Stokes’ Theorem Let 􀀨 be a piecewise smooth oriented surface with a boundary that is a simple closed curve 􀀘 with positive orientation. If ®􀀛 is a vector field with component functions that have continuous partial derivatives on an open region containing 􀀨, Then ∫􀀘 ®􀀛 · 􀀳􀁁® = ∫ ∫􀀨 (􀀲􀁄􀁁􀀻®􀀛) · 􀀳®􀀨 ■ The Divergence Theorem Let 􀀚 be a simple solid region and let 􀀨 be the boundary surface of 􀀚, given with positive orientation. Let ®􀀛 be a vector field

FAQs

What is the Second Derivative Test for a function of two variables?

Let f(x,y) have continuous first- and second-order partial derivatives near a critical point (x0,y0). Define D=fxx(x0,y0)fyy(x0,y0)−(fxy(x0,y0))^2: D>0 with fxx>0 gives a local minimum, D>0 with fxx<0 gives a local maximum, D<0 gives a saddle point, and D=0 is inconclusive.

When is a vector field conservative in R^3?

If a vector field F is defined on all of R^3, its component functions have continuous partial derivatives, and curl F = 0, then F is conservative. In that case, div(curl F)=0 is also stated.

Which theorems relate line, surface, and volume integrals in the document?

The document states the Fundamental Theorem of Line Integrals, Green’s Theorem (circulation form) for planar regions, Stokes’ Theorem for oriented surfaces with boundary curves, and the Divergence Theorem connecting flux through a closed surface to a triple integral over the enclosed solid.

Nov 1, 2026

Uploaded by RizkyPratama_92