ParametricPlot[{t,t^2},{t,-5,5}]

ParametricPlot[{Cos[t],Sin[t]},{t,0,Pi}]

ParametricPlot[{Sin[t],Cos[t]},{t,0,Pi}]

ParametricPlot3D[{5Cos[t],5Sin[t],t},{t,0,6 Pi}]

ParametricPlot3D[{1+2t,3-t,5+3t},{t,-5,5}]

ParametricPlot[{3+2Cos[t],2+2Sin[t]},{t,0,3Pi/2}]

ParametricPlot[{3+2Sin[t- Pi/2],2+2Cos[t- Pi/2]},{t,0,3Pi/2}]

sphere=ContourPlot3D[x^2+y^2+z^2==81,{x,-10,10},{y,-10,10},{z,-10,10}];
line=ParametricPlot3D[{t,2t,2+3},{t,-5,5},PlotStyle->{Black,Thickness[.02]}];
Show[line,sphere]

ParametricPlot3D[{{t,1+2t,3-2t},{-2-2t,1-2t,1+t}},{t,-5,5}]

ParametricPlot3D[{{t-1,1+2t,5-t},{2+2t,4+t,3+t}},{t,-5,5}]

circle=ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi}];
arrows=Graphics[{Thickness[.01],Arrow[{{1,0},{1,1}}],Arrow[{{Sqrt[2]/2,Sqrt[2]/2},{0,Sqrt[2]}}]}];
points=Graphics[{PointSize[.04],Point[{1,0}],Point[{Sqrt[2]/2,Sqrt[2]/2}]}];
Show[circle,arrows,points,PlotRange->{{-1,1},{-1,2}}]

graph=ParametricPlot3D[{t^2,t^3,2t},{t,0,3}];
arrows=Graphics3D[{Thickness[.01],Arrow[{{4,8,4},{8,20,6}}]}];
points=Graphics3D[{PointSize[.04],Point[{4,8,4}]}];
Show[graph,arrows,points,PlotRange->Automatic]

circle=ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi}];
arrows=Graphics[{Thickness[.01],Arrow[{{1,0},{0,0}}],Arrow[{{Sqrt[2]/2,Sqrt[2]/2},{0,0}}]}];
points=Graphics[{PointSize[.04],Point[{1,0}],Point[{Sqrt[2]/2,Sqrt[2]/2}]}];
Show[circle,arrows,points,PlotRange->{{-1,1},{-1,2}}]

cylinder=ParametricPlot3D[{s,3Cos[t]+3,3Sin[t]-4},{s,-5,5},{t,0,2Pi}];
circle=ParametricPlot3D[{0,3Cos[t]+3,3Sin[t]-4},{t,0,2Pi},PlotStyle->{Black,Thickness[.02]}];
helix=ParametricPlot3D[{t, 3Cos[t]+3,3Sin[t]-4},{t,0,2Pi},PlotStyle->{Blue,Thickness[.02]}];
Show[cylinder,circle,helix]

ParametricPlot3D[{10/Sqrt[y] Cos[t],y,10/Sqrt[y]Sin[t]},{y,0,20},{t,0,2 Pi}]

RegionPlot[x^2/9 + y^2/25 ≤1,{x,-5,5},{y,-5,5}]

RegionPlot[Abs[y]≤x&&Abs[y]≤5-x,{x,0,5},{y,-3,3}]

VectorPlot[{-y,x},{x,-3,3},{y,-3,3}]

VectorPlot[{x,0},{x,-3,3},{y,-3,3}]

VectorPlot[{y,0},{x,-3,3},{y,-3,3}]

VectorPlot[{-x,-y},{x,-3,3},{y,-3,3}]

VectorPlot[-#/Norm[#]^3&[{x,y}],{x,-1,1},{y,-1,1},VectorPoints->20,VectorScale->.3,RegionFunction->(Norm[{#,#2}]>.1&),ImageSize->500,PlotRange->1]

VectorPlot3D[-#/Norm[#]^3&[{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1},VectorPoints->20,VectorScale->.3,RegionFunction->(Norm[{#,#2,#3}]>.1&),ImageSize->500,PlotRange->1]

VectorPlot3D[-#/Norm[#]^3&[{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1},VectorPoints->20,VectorScale->.6,RegionFunction->(Norm[{#,#2,#3}]>.1&),ImageSize->500,PlotRange->1]

VectorPlot3D[{-y,x,z},{x,-3,3},{y,-3,3},{z,-3,3}]

VectorPlot[{y,x},{x,-3,3},{y,-3,3}]

ellipsecontour=ContourPlot[{3x^2+y^2==1,3x^2+y^2==4,3x^2+y^2==9,3x^2+y^2==16},{x,-4,4},{y,-4,4}]
ellipsegradient=VectorPlot[{6x,2y},{x,-4,4},{y,-4,4}]
Show[ellipsecontour,ellipsegradient]

StreamPlot[{5,-1},{x,-3,3},{y,-3,3}]

StreamPlot[{-y,x},{x,-3,3},{y,-3,3}]

plot=StreamPlot[{x^2+y,x+y^2},{x,-3,3},{y,-3,3}]
points=Graphics[{PointSize[.03],Point[{{-2,1},{3,0}}]}];
points2=Graphics[{PointSize[.03],Point[{{-2,1},{1/2,1/2},{7/8,7/8}}]}];
Show[plot,points]
Show[plot,points2]
points4=Graphics[{PointSize[.03],Point[{{-2.`,1.`},{-0.75`,0.75`},{-0.421875`,0.703125`},{-0.20159912109375`,0.72125244140625`},{-0.011125459335744381`,0.8009039321914315`}}]}];
Show[plot,points4]
points20=Graphics[{PointSize[.03],Point[{{-2.`,1.`},{-1.75`,0.95`},{-1.549375`,0.9076249999999999`},{-1.38396560546875`,0.8713454070312499`},{-1.2446302952611632`,0.8401092676755351`},{-1.1251696032832919`,0.8131669319941931`},{-1.0212109248759482`,0.7899704747944709`},{-0.9295688134819252`,0.7701125961030236`},{-0.8478582747268644`,0.7532878259627543`},{-0.774250700727586`,0.7392670396635957`},{-0.7073141413655483`,0.7278802924238653`},{-0.6459054620155709`,0.7190050713605405`},{-0.5890955151544665`,0.7125582128918707`},{-0.5361159282111176`,0.7084903974721302`},{-0.48632039391342774`,0.7067825332270852`},{-0.4391558909752679`,0.7074435909951585`},{-0.39414081659659594`,0.7105096181684014`},{-0.3508479865228043`,0.716043773214062`},{-0.3088910823797459`,0.7241373081458533`},{-0.26791353193376666`,0.7349114960793022`},{-0.22757907410014028`,0.7485205648360898`}}]}];
Show[plot,points20]

eulerplot[df_,xinit_,yinit_,xfin_,n_]:=(r[0]=N[yinit];stepsize=(xfin - xinit)/n;Do[r[k+1]=r[k]+(df/.x->(First[r[k]])/.y->Last[r[k]])*stepsize,{k,0,n-1}];
points=Table[r[k],{k,0,n}];
ListPlot[points,PlotMarkers->{Automatic,Small},PlotStyle->{Black}])

eulerplot[{x^2+y,x+y^2},0,{-2,1},1,20]
points