This happened to be some of what I needed to know this speci64257c semester in my course For example Stokes Theorem is not even mentioned 2 Vectors Between Two Points Given y y PQ 3 Vectors in the Plane let 31 Simple Operations cv cv brPage 2br 32 ID: 25920 Download Pdf

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This happened to be some of what I needed to know this speci64257c semester in my course For example Stokes Theorem is not even mentioned 2 Vectors Between Two Points Given y y PQ 3 Vectors in the Plane let 31 Simple Operations cv cv brPage 2br 32

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Multivariable Calculus Study Guide: A L X Version Tyler Silber University of Connecticut December 11, 2011 1 Disclaimer It is not guaranteed that I have every single bit of necessary information for the course. This happened to be some of what I needed to know this speciﬁc semester in my course. For example, Stokes’ Theorem is not even mentioned. 2 Vectors Between Two Points Given ,y ) & ,y PQ 3 Vectors in the Plane let 3.1 Simple Operations cv cv

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3.2 Unit Vectors 3.3 Vectors of a Speciﬁed Length 4 Vectors in Three Dimensions 4.1 Notes Everything

in the above section can be expanded to three dimensions. Simply add another component. 4.2 Random Equations xy -plane x,y,z ) : = 0 xz -plane x,y,z ) : = 0 yz -plane x,y,z ) : = 0 Sphere: ( + ( + ( 5 Dot Product 5.1 Deﬁnitions || cos = 0 ±| ||

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5.2 Projections The orthogonal projection of onto is denoted proj and the scalar compo- nent of in the direction of is denoted scal proj cos scal cos 6 Cross Product || sin (1) Note: is orthogonal to both and and the direction is deﬁned by the right-hand rule. 7 Lines and Curves in Space 7.1 Vector-Valued Functions ) = ,y

,z 7.2 Lines x,y,z ,y ,z a,b,c for 7.3 Line Segments Given ,y ,z ) & ,y ,z ,y ,z ,y ,z for 0 7.4 Curves in Space ) = ,g ,h Equation 1 is also equal to the area of the parallelogram created by the two vectors.

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7.5 Limits lim ) = lim lim lim 8 Calculus of Vector-Valued Functions 8.1 Derivative and Tangent Vector ) = Note: ) is the tangent vector to ) at the point ( ,g ,h )). 8.2 Indeﬁnite Integral dt ) + Note: is an arbitrary constant vector and 8.3 Deﬁnite Integral dt dt dt dt 9 Motion in Space 9.1 Deﬁnitions ) = ) = 00 Speed 9.2 Two-Dimensional Motion in a

Gravitational Field Given (0) = ,v (0) = ,y ) = ,y gt ) = ,y gt

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9.3 Two-Dimensional Motion Given (0) = 〈| cos θ, sin (0) = Time sin Range sin 2 MaxHeight sin 10 Planes and Surfaces 10.1 Plane Equations The plane passing through the point ,y ,z ) with a normal vector a,b,c, is described by the equations: ) + ) + ) = 0 ax by cz d, where ax by cz In order to ﬁnd the equation of a plane when given three points, simply create any two vectors out of the points and take the cross product to ﬁnd the vector normal to the plane. Then use one of the above

formulae. 10.2 Parallel and Orthogonal Planes Two planes are parallel if their normal vectors are parallel. Two planes are orthogonal if their normal vectors are orthogonal. 10.3 Surfaces 10.3.1 Ellipsoid = 1 10.3.2 Elliptic Paraboloid It would be worth it to learn how to derive sections 9.2 and 9.3.

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10.3.3 Hyperboloid of One Sheet = 1 10.3.4 Hyperboloid of Two Sheets = 1 10.3.5 Elliptic Cone 10.3.6 Hyperbolic Paraboloid 11 Graphs and Level Curves 11.1 Functions of Two Variables x,y x,y,z ) = 0 11.2 Functions of Three Variables x,y,z w,x,y,z ) = 0 11.3 Level Curves Imagine

stepping onto a surface and walking along a path with constant eleva- tion. The path you walk on is known as the contour curve, while the projection of the path onto the xy -plane is known as a level curve.

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12 Limits and Continuity 12.1 Limits The function has the limit as x,y ) approaches a,b ). lim x,y a,b x,y ) = lim x,y ) = If x,y ) approaches two diﬀerent values as ( x,y ) approaches ( a,b ) along two diﬀerent paths in the domain of , then the limit does not exist. 12.2 Continuity The function if continuous at the point ( a,b ) provided: lim x,y a,b x,y ) =

a,b 13 Partial Derivatives 13.1 Deﬁnitions a,b ) = lim h,b a,b a,b ) = lim a,b a,b So basically just take the derivative of one (the subscript) given that the other one is a constant. 13.2 Notation for Higher-Order Partial Derivatives ∂x ∂f ∂x ∂x = ( xx ∂y ∂f ∂y ∂y = ( yy ∂x ∂f ∂y ∂x∂y = ( yx ∂y ∂f ∂x ∂y∂x = ( xy Note: xy yx for nice functions. 13.3 Diﬀerentiability Suppose the function has partial derivatives and deﬁned on an open region containing ( a,b ), with and

continuous at ( a,b ). Then is diﬀeren- tiable at ( a,b ). This also implies that it is continuous at ( a,b ).

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14 Chain Rule 14.1 Examples You can use a tree diagram to determine the equation for the chain rule. You can also just think about it. Refer to the following examples. is a function of and , while and are functions of dz dt ∂z ∂x dx dt ∂z ∂y dy dt is a function of , and , while , and are functions of dw dt ∂w ∂x dx dt ∂w ∂y dy dt ∂w ∂z dz dt is a function of and , while and are functions of and

∂z ∂s ∂z ∂x ∂x ∂s ∂z ∂y ∂y ∂s is a function of is a function of and and are functions of dw dt dw dz ∂z ∂x dx dt ∂z ∂y dy dt 14.2 Implicit Diﬀerentiation Let be diﬀerentiable on its domain and suppose that x,y ) = 0 deﬁnes as a diﬀerentiable function of . Provided = 0, dy dx 15 Directional Derivatives and Gradient 15.1 Deﬁnitions Let be diﬀerentiable at ( a,b ) and let ,u be a unit vector in the xy -plane. The directional derivative of at ( a,b ) in the direction of is

a,b ) = a,b ,f a,b 〉· ,u a,b Gradient x,y ) = x,y ,f x,y x,y x,y

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15.2 Directions of Change has its maximum rate of increase at ( a,b ) in the direction of the gradient a,b ). The rate of increase in this direction is | a,b has its maximum rate of decrease at ( a,b ) in the direction of the gradient a,b ). The rate of decrease in this direction is −| a,b The directional derivative is zero in any direction orthogonal to a,b ). 15.3 Expanding to Three Dimensions It’s really intuitive how it expands into three dimensions. Just add another component or where you think

it should go. 16 Tangent Plane and Linear Approximation 16.1 Tangent Plane for F ) = The tangent plane passes through the point a,b,c ). a,b,c )( ) + a,b,c )( ) + a,b,c )( ) = 0 16.2 Tangent Plane for z The tangent plane passes through the point ( a,b,f a,b )). a,b )( ) + a,b )( ) + a,b 16.3 Linear Approximation Firstly, calculate the equation of the tangent plane of a point near the point you wish to approximate. Then simply plug in the point and you’re done. 16.4 The diﬀerential dz The change in x,y ) as the independent variables change from ( a,b ) to dx,b dy ) is denoted and is

approximated by the diﬀerential dz dz a,b dx a,b dy 17 Max-Min Problems 17.1 Derivatives and Local Maximum/Minimum Values If has a local maximum or minimum value at ( a,b ) and the partial derivatives and exist at ( a,b ), then a,b ) = a,b ) = 0.

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17.2 Critical Points A critical point exists if either a,b ) = a,b ) = 0 one (or both) of or does not exist at ( a,b 17.3 Second Derivative Test Let x,y ) = xx yy xy If a,b 0 and xx a,b 0, then has a local maximum at ( a,b ). If a,b 0 and xx a,b 0, then has a local minimum at ( a,b ). If a,b 0, then has a saddle point at ( a,b

). If a,b ) = 0, then the test is inconclusive. 17.4 Absolute Maximum/Minimum Values Let be continuous on a closed bounded set in . To ﬁnd absolute maximum and minimum values of on 1. Determine the values of at all critical points in 2. Find the maximum and minimum values of on the boundary of 3. The greatest function value found in Steps 1 and 2 is the absolute maxi- mum value of on , and the least function value found in Steps 1 and 2 is the absolute minimum values of on 18 Double Integrals 18.1 Double Integrals on Rectangular Regions Let be continuous on the rectangular region x,y )

: b,c . The double integral of over may be evaluated by either of two iterated integrals: ZZ x,y dA x,y dxdy x,y dydx 10

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18.2 Double Integrals over Nonrectangular Regions Let be a region bounded below and above by the graphs of the continuous functions ) and ), respectively, and by the lines and If is continuous on , then ZZ x,y dA x,y dydx Let be a region bounded on the left and right by the graphs of the continuous functions ) and ), respectively, and by the lines and If is continuous on , then ZZ x,y dA x,y dxdy 18.3 Areas of Regions by Double Integrals area of ZZ dA 19

Polar Double Integrals 19.1 Double Integrals over Polar Rectangular Regions Let be continuous on the region in the xy -plane r, ) : 0 b, , where . Then ZZ r, dA r, rdrd 19.2 Double Integrals over More General Polar Regions Let be continuous on the region in the xy -plane r, ) : 0 , where . Then. ZZ r, dA r, rdrd If is nonnegative on , the double integral gives the volume of the solid bounded by the surface r, ) and 11

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19.3 Area of Polar Regions ZZ dA rdrd 20 Triple Integrals Let x,y,z ) : b,g ,G x,y x,y , where are continuous functions. The triple integral of a continuous

function on is evaluated as the iterated integral ZZZ x,y,z dV x,y x,y x,y,z dzdydx 21 Cylindrical and Spherical Coordinates 21.1 Deﬁnitions 21.1.1 Cylindrical Coordinates r,θ,z ) An extension of polar coordinates into . Simply add a component. 21.1.2 Spherical Coordinates ρ,ϕ, is the distance from the origin to a point is the angle between the positive -axis and the line OP is the same angle as in cylindrical coordinates; it measure rotation about the -axis relative to the positive -axis. 21.2 Rectangular to Cylindrical tan 21.3 Cylindrical to Rectangular cos sin 12

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21.4 Integration in Cylindrical Coordinates ZZZ r,θ,z dV cos θ,r sin cos θ,r sin r,θ,z dzrdrd 21.5 Rectangular to Spherical You have to solve for and with trigonometry. 21.6 Spherical to Rectangular sin cos sin sin cos 21.7 Integration in Spherical Coordinates ZZZ ρ,ϕ, dV ϕ, ϕ, ρ,ϕ, sin ϕdρdϕd 22 Change of Variables 22.1 Jacobian Determinant of a Transformation of Two Variables Given a transformation u,v ,y u,v ), where and are diﬀeren- tiable on a region of the uv -plane, the Jacobian determinant of is

u,v ) = x,y u,v ∂x ∂u ∂x ∂v ∂y ∂u ∂y ∂v 22.2 Change of Variables for Double Integrals ZZ x,y dA ZZ u,v ,h u,v )) u,v dA 22.3 Change of Variables for Triple Integrals I am SO not typing out the expansion of the above into triple integrals. It’s intuitive. Just add stuﬀ where you think it should go. 13

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22.4 YOU have to Choose the Transformation Just cry. 23 Vector Fields 23.1 Vector Fields in Two Dimensions x,y ) = x,y ,g x,y 23.2 Radial Vector Fields in R Let = ( x,y ). A vector ﬁeld of the form x,y , where is a

scalar-valued function, is a radial vector ﬁeld. x,y ) = x,y is a real number. At every point (sans origin), the vectors of this ﬁeld are directed outward format he origin with a magnitude of . You can also apply all of this to by just adding a component. 23.3 Gradient Fields and Potential Functions Let x,y ) and x,y,z ) be diﬀerentiable functions on regions of and , respectively. The vector ﬁeld is a gradient ﬁeld, and the function is a potential function for 24 Line Integrals 24.1 Evaluating Scalar Line Integrals in R Let be continuous on a region

containing a smooth curve ) = ,y for . Then fds ,y )) dt ,y )) dt 24.2 Evaluating Scalar Line Integrals in R Simply add a component to the above where it obviously belongs. 14

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24.3 Line Integrals of Vector Fields 24.3.1 Deﬁnition Let be a vector ﬁeld that is continuous on a region containing a smooth oriented curve parametrized by arc length. Let be the unit tangent vector at each point of consistent with the orientation. The line integral of over is ds 24.3.2 Diﬀerent Forms f,g,h and has a parametrization ) = ,y ,z , for dt fx )+ gy )+ hz )) dt fdx gdy

hdz For line integrals in the plane, we let f,g and assume is parametrized in the form ) = ,y , for . Then ds fx ) + gy )) dt fdx gdy 24.4 Work is a force ﬁeld ds dt 24.5 Circulation is a vector ﬁeld Circulation ds 24.6 Flux Flux ds fy gx )) dt , and a positive answer means a positive outward ﬂux. 25 Conservative Vector Fields 25.1 Test for Conservative Vector Field Let f,g,h be a vector ﬁeld deﬁned on a connected and simply connected region of , where , and have continuous ﬁrst partial derivatives on 15

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. Then, is a conservative

vector ﬁeld on (there is a potential function such that ) if and only if ∂f ∂y ∂g ∂x ∂f ∂z ∂h ∂x ∂g ∂z ∂h ∂y For vector ﬁelds in , we have the single condition ∂f ∂y ∂g ∂x 25.2 Finding Potential Functions Suppose f,g,h is a conservative vector ﬁeld. To ﬁnd such that take the following steps: 1. Integrate with respect to to obtain , which includes an arbitrary function y,z 2. Compute and equate it to to obtain an expression for y,z ). 3. Integrate y,z ) with respect to to

obtain y,z ), including an arbitrary function ). 4. Compute and equate it to to get ). Beginning the procedure with or may be easier in some cases. This method can also be used to check if a vector ﬁeld is conservative by seeing if there is a potential function. 25.3 Fundamental Theorem for Line Integrals ds 25.4 Line Integrals on Closed Curves Let in (or in ) be an open region. Then is a conservative vector ﬁeld on if and only if = 0 on all simple closed smooth oriented curves in 26 Green’s Theorem 26.1 Circulation Form fdx gdy ZZ ∂g ∂x ∂f ∂y dA 16

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26.2 Area of a Plane Region by Line Integrals xdy ydx xdy ydx 26.3 Flux Form ds fdy gdx ZZ ∂f ∂x ∂g ∂y dA 27 Divergence and Curl 27.1 Divergence of a Vector Field div( ) = ∇· ∂f ∂x ∂g ∂y ∂h ∂z 27.2 Divergence of Radial Vector Fields div( ) = x,y,z p/ 27.3 Curl curl( ) = ∇× Just derive the curl by doing the cross product. 27.4 Divergence of the Curl ∇· ∇× ) = 0 28 Surface Integrals 28.1 Parameterization 28.1.1 z is Explicitly Deﬁned Use , and since is explicitly deﬁned, you

already have what equals. 28.1.2 Cylinder Simply use cylindrical coordinates to parameterize the surface in terms of and 17

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28.1.3 Sphere Simply use spherical coordinates to parameterize the surface in terms of and 28.1.4 Cone Use: cos sin and 0 28.2 Surface Integrals of Parameterized Surfaces ZZ x,y,z d ZZ u,v ,y u,v ,z u,v )) ∂s ∂t dA 29 Divergence Theorem Let be a vector ﬁeld whose components have continuous ﬁrst partial deriva- tives in a connected and simply connected region enclosed by a smooth ori- ented surface . Then ZZ dS ZZZ ∇· dV

where is the outward normal vector on 18

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