fzus63 kdtx 120803
glflh 
expires:202608121615;;045760

open lake forecast for lake huron
national weather service detroit/pontiac mi
403 am edt wed aug 12 2026

for waters beyond five nautical miles off shore on lake huron

waves are the significant wave height - the average of the highest
1/3 of the wave spectrum. occasional wave height is the average of
the highest 1/10 of the wave spectrum.

.synopsis...a quasi-stationary front holds over the southern great
lakes through thursday, bringing waves of showers and
thunderstorms across the waters. high pressure, 30.00 inches,
will dominate the region to the north of the front. light winds
will be variable over the next 2 days, but maintain an overall
northerly trend over lake huron and southerly toward western lake
erie.


lhz361-121615-
lake huron from 5nm east of mackinac bridge to presque isle lt
beyond 5 nm off shore-
403 am edt wed aug 12 2026

.today...light and variable winds. waves 2 feet or less. 
.tonight...northwest winds 5 to 10 knots. waves 2 feet or less. 
.thursday...northwest winds 5 to 10 knots early in the morning
becoming light and variable. waves 2 feet or less. 
.thursday night...northwest winds 5 to 10 knots. waves 2 feet or
less. 
.friday...north winds 5 to 10 knots becoming light and variable.
waves 2 feet or less. 
.saturday...southeast winds 5 to 10 knots increasing to 10 to
15 knots. showers likely and a chance of thunderstorms by
midnight. waves 2 feet or less building to 2 to 4 feet . 
.sunday...south winds 10 to 15 knots veering to the northwest .
showers likely and a chance of thunderstorms early in the
morning...then a chance of showers in the late morning and early
afternoon. waves 1 to 3 feet. 


$$

fzus63 kdtx 120803
glflh 
expires:202608121615;;045761

open lake forecast for lake huron
national weather service detroit/pontiac mi
403 am edt wed aug 12 2026

for waters beyond five nautical miles off shore on lake huron

waves are the significant wave height - the average of the highest
1/3 of the wave spectrum. occasional wave height is the average of
the highest 1/10 of the wave spectrum.

.synopsis...a quasi-stationary front holds over the southern great
lakes through thursday, bringing waves of showers and
thunderstorms across the waters. high pressure, 30.00 inches,
will dominate the region to the north of the front. light winds
will be variable over the next 2 days, but maintain an overall
northerly trend over lake huron and southerly toward western lake
erie.


lhz362-363-121615-
lake huron from presque isle light to sturgeon point mi beyond
5nm off shore-
lake huron from sturgeon point to alabaster mi beyond 5nm off
shore-
403 am edt wed aug 12 2026

.today...light and variable winds. a chance of showers in the
afternoon. waves 2 feet or less. 
.tonight...light and variable winds becoming northwest 5 to
10 knots after midnight. waves 2 feet or less. 
.thursday...northwest winds 5 to 10 knots early in the morning
becoming light and variable. waves 2 feet or less. 
.thursday night...light and variable winds becoming northwest
5 to 10 knots after midnight. waves 2 feet or less. 
.friday...north winds 5 to 10 knots becoming light and variable.
waves 2 feet or less. 
.saturday...east winds 5 to 10 knots becoming southeast 10 to
15 knots by midnight. showers likely and a chance of
thunderstorms by midnight. waves 2 feet or less. 
.sunday...south winds 10 to 15 knots veering to the west early in
the evening...then veering to the west 10 to 15 knots by
midnight. showers likely and a chance of thunderstorms early in
the morning...then a chance of showers in the late morning and
early afternoon. waves 1 to 3 feet. 


$$

fzus63 kdtx 120803
glflh 
expires:202608121615;;045762

open lake forecast for lake huron
national weather service detroit/pontiac mi
403 am edt wed aug 12 2026

for waters beyond five nautical miles off shore on lake huron

waves are the significant wave height - the average of the highest
1/3 of the wave spectrum. occasional wave height is the average of
the highest 1/10 of the wave spectrum.

.synopsis...a quasi-stationary front holds over the southern great
lakes through thursday, bringing waves of showers and
thunderstorms across the waters. high pressure, 30.00 inches,
will dominate the region to the north of the front. light winds
will be variable over the next 2 days, but maintain an overall
northerly trend over lake huron and southerly toward western lake
erie.


lhz462>464-121615-
lake huron from port austin to harbor beach beyond 5nm off shore-
lake huron from harbor beach to port sanilac beyond 5nm off shore-
lake huron from port sanilac to port huron beyond 5nm off shore-
403 am edt wed aug 12 2026

.today...south winds 5 to 10 knots early in the morning becoming
light and variable...then becoming north 5 to 10 knots late in
the afternoon. patchy fog in the morning. waves 2 feet or less. 
.tonight...north winds 5 to 10 knots early in the evening
becoming light and variable. waves 2 feet or less. 
.thursday...light and variable winds becoming north 5 to 10 knots
early in the afternoon. waves 2 feet or less. 
.thursday night...light and variable winds becoming northeast
5 to 10 knots after midnight. waves 2 feet or less. 
.friday...north winds 5 to 10 knots increasing to 10 to 15 knots
late in the afternoon...then becoming northeast 5 to 10 knots
late in the evening. waves 2 feet or less building to 2 to 4 feet
late in the afternoon. 
.saturday...east winds 5 to 10 knots becoming southeast 10 to
15 knots by midnight. showers likely and a chance of
thunderstorms by midnight. waves 1 to 3 feet. 
.sunday...south winds 10 to 15 knots veering to the west early in
the evening...then veering to the west 10 to 15 knots by
midnight. showers likely until afternoon. a chance of
thunderstorms. waves 1 to 3 feet. 


$$

fzus63 kmkx 120758
glflm 
expires:202608121515;;045443

open lake forecast for lake michigan
national weather service milwaukee/sullivan wi
258 am cdt wed aug 12 2026

for waters beyond five nautical miles of shore on lake michigan.

waves are provided as a range of significant wave heights, which
is the average of the highest 1/3 of the wave, along with the
average height of the highest 10% of the waves which will
occasionally be encountered.

.synopsis...

no major changes to the going forecast this morning.

light and variable winds are expected to continue through 
thursday as a stalled frontal boundary remains south of the lake 
bringing periodic shower and thunderstorm chances for southern 
portions of lake michigan along with patchy fog. influence from 
this front finally moves away from the region as high pressure 
around around 30.3 inches builds down across the great lakes 
region thursday night into friday. will see south to southeasterly
winds increase into the weekend as the high pressure slides east 
of the area while low pressure develops over the plains. expect 
this low pressure to track northeastward across the upper great 
lakes later saturday into sunday bringing shower and thunderstorm
chances to the lake. behind this departing low pressure, winds
shift to the northwest later sunday into early next week as
another high pressure builds down across the region.


lmz261-362-364-366-563-565-567-868-121515-
lake michigan from seul choix point to rock island passage 5 nm
offshore to mid lake-
lake michigan south of a line from seul choix point to the
mackinac bridge and north of a line from charlevoix mi to south
fox island 5 nm offshore-
lake michigan from charlevoix to point betsie mi 5 nm offshore to
mid lake-
lake michigan from point betsie to manistee mi 5 nm offshore to
mid lake-
lake michigan from rock island passage to sturgeon bay wi-
lake michigan from sturgeon bay to two rivers wi 5 nm offshore to
mid lake-
lake michigan from two rivers to sheboygan wi 5 nm offshore to
mid lake-
lake michigan from pentwater to manistee mi 5 nm offshore to mid
lake-
258 am cdt wed aug 12 2026

sheboygan wi to pentwater mi north...

.today...east winds less than 10 kt becoming northeast late this
morning, then becoming north early this afternoon increasing to
10 to 15 kt late. patchy fog this morning. waves 1 ft or less. 
.tonight...north winds 5 to 10 kt becoming northeast. waves 1 ft
or less. 
.thursday...northeast winds 5 to 10 kt increasing to 10 to 15 kt.
waves 1 ft or less. 
.thursday night...north winds 10 to 20 kt becoming northeast.
waves 1 to 2 ft building to 1 to 3 ft. 
.friday...northeast winds 10 to 20 kt increasing to 15 to 25 kt.
waves 2 to 4 ft. 
.friday night...northeast winds 15 to 25 kt becoming southeast.
waves 2 to 4 ft. 
.saturday...southeast winds to 30 kt. chance of showers. waves
4 to 6 ft occasionally to 8 ft. 
.sunday...northwest winds 10 to 20 kt becoming north 15 to 25 kt.
waves 2 to 4 ft.  

$$

fzus63 kmkx 120758
glflm 
expires:202608121515;;045444

open lake forecast for lake michigan
national weather service milwaukee/sullivan wi
258 am cdt wed aug 12 2026

for waters beyond five nautical miles of shore on lake michigan.

waves are provided as a range of significant wave heights, which
is the average of the highest 1/3 of the wave, along with the
average height of the highest 10% of the waves which will
occasionally be encountered.

.synopsis...

no major changes to the going forecast this morning.

light and variable winds are expected to continue through 
thursday as a stalled frontal boundary remains south of the lake 
bringing periodic shower and thunderstorm chances for southern 
portions of lake michigan along with patchy fog. influence from 
this front finally moves away from the region as high pressure 
around around 30.3 inches builds down across the great lakes 
region thursday night into friday. will see south to southeasterly
winds increase into the weekend as the high pressure slides east 
of the area while low pressure develops over the plains. expect 
this low pressure to track northeastward across the upper great 
lakes later saturday into sunday bringing shower and thunderstorm
chances to the lake. behind this departing low pressure, winds
shift to the northwest later sunday into early next week as
another high pressure builds down across the region.


lmz080-669-671-673-675-777-779-870-872-874-876-878-121515-
lake michigan from michigan city in to st. joseph mi 5 nm
offshore to mid lake-
lake michigan from sheboygan to port washington wi 5 nm offshore
to mid lake-
lake michigan from port washington to north point light wi 5 nm
offshore to mid lake-
lake michigan from north point light to wind point wi 5 nm
offshore to mid lake-
lake michigan from wind point wi to winthrop harbor il 5 nm
offshore to mid lake-
lake michigan from winthrop harbor to wilmette harbor il 5 nm
offshore to mid lake-
lake michigan from wilmette harbor il to michigan city in 5 nm
offshore to mid lake-
lake michigan from whitehall to pentwater mi 5 nm offshore to mid
lake-
lake michigan from grand haven to whitehall mi 5 nm offshore to
mid lake-
lake michigan from holland to grand haven mi 5 nm offshore to mid
lake-
lake michigan from south haven to holland mi 5 nm offshore to mid
lake-
lake michigan from st. joseph to south haven mi 5 nm offshore to
mid lake-
258 am cdt wed aug 12 2026

sheboygan wi to pentwater mi south...

.today...southeast winds 5 to 10 kt backing to north. patchy fog
this morning. waves nearly calm. 
.tonight...north winds 5 to 10 kt becoming east. waves nearly
calm. 
.thursday...east winds 5 to 10 kt becoming northeast. waves 1 ft
or less. 
.thursday night...northeast winds 10 to 20 kt. chance of showers
in the evening. waves 1 to 2 ft building to 1 to 3 ft. 
.friday...northeast winds 10 to 20 kt increasing to 15 to 25 kt.
waves 2 to 4 ft. 
.friday night...northeast winds 10 to 20 kt becoming southeast
15 to 25 kt. waves 3 to 5 ft. 
.saturday...southeast winds to 30 kt. chance of showers. waves
4 to 6 ft occasionally to 8 ft. 
.sunday...west winds 10 to 20 kt becoming northwest 15 to 25 kt.
chance of showers and thunderstorms. waves 2 to 4 ft. 

$$

fzus63 kmqt 120801
glfls 
expires:202608122000;;045624

open lake forecast for lake superior
national weather service marquette mi
401 am edt wed aug 12 2026

for waters beyond 5 nautical miles of shore on lake superior.

waves are the significant wave height - the average of the
highest 1/3 of the wave spectrum. occasional wave height is the
average of the highest 1/10 of the wave spectrum.

.synopsis...

this morning, 29.5 inch low pressure is over the gulf of st lawrence 
and deepens to 29.5 inches while a plains low of 29.7 inches is only 
separated from the first low by weak 29.8-29.9 inch ridging over 
lake superior. this pattern holds until thursday morning, when a 
30.2 inch high over the canadian rockies begins to extend stronger 
ridging over lake superior before eventually becoming centered over 
the lake at 30.2 inches friday afternoon. a 29.9 inch trough then 
moves over the western end of the lake saturday afternoon as the 
30.2 inch high moves over the lower great lakes. this trough 
consolidates to a 29.8 inch low as it passes over lake superior, 
reaching ontario by sunday afternoon, with 30.1 inch ridging to 
follow.




lsz162-122000-
lake superior west of a line from saxon harbor wi to grand
portage mn beyond 5nm-
401 am edt wed aug 12 2026 /301 am cdt wed aug 12 2026/

.early this morning...north winds less than 10 knots. patchy fog.
waves calm to 2 feet. 
.today...northwest winds less than 10 knots becoming south 5 to
15 knots in the afternoon. patchy dense fog through early
afternoon. waves calm to 2 feet. 
.tonight...northeast winds 5 to 15 knots backing north after
midnight, then veering northeast late. a chance of showers. waves
calm to 2 feet. 
.thursday...northeast winds 5 to 15 knots veering east late.
waves calm to 2 feet. 
.thursday night...northeast winds 5 to 15 knots. waves calm to
2 feet. 
.friday...northeast winds 10 to 20 knots increasing to 15 to
25 knots by mid afternoon. waves building to 2 to 4 feet, then
subsiding to 1 to 3 feet. 
.friday night...northeast winds 15 to 25 knots diminishing to
5 to 15 knots after midnight. waves subsiding to calm to 2 feet. 
.saturday...northeast winds 5 to 15 knots backing north. a chance
of showers. waves calm to 2 feet. 
.sunday...west winds 5 to 15 knots veering northwest. waves calm
to 2 feet. 

$$

fzus63 kmqt 120801
glfls 
expires:202608122000;;045625

open lake forecast for lake superior
national weather service marquette mi
401 am edt wed aug 12 2026

for waters beyond 5 nautical miles of shore on lake superior.

waves are the significant wave height - the average of the
highest 1/3 of the wave spectrum. occasional wave height is the
average of the highest 1/10 of the wave spectrum.

.synopsis...

this morning, 29.5 inch low pressure is over the gulf of st lawrence 
and deepens to 29.5 inches while a plains low of 29.7 inches is only 
separated from the first low by weak 29.8-29.9 inch ridging over 
lake superior. this pattern holds until thursday morning, when a 
30.2 inch high over the canadian rockies begins to extend stronger 
ridging over lake superior before eventually becoming centered over 
the lake at 30.2 inches friday afternoon. a 29.9 inch trough then 
moves over the western end of the lake saturday afternoon as the 
30.2 inch high moves over the lower great lakes. this trough 
consolidates to a 29.8 inch low as it passes over lake superior, 
reaching ontario by sunday afternoon, with 30.1 inch ridging to 
follow.




lsz263-122000-
lake superior from saxon harbor wi to upper entrance to portage
canal mi 5nm off shore to the us/canadian border including isle
royale national park-
401 am edt wed aug 12 2026

.early this morning...northwest winds less than 10 knots. waves
calm to 2 feet. 
.today...northwest winds less than 10 knots increasing to 5 to
15 knots in the afternoon, then backing south late. waves calm to
2 feet. 
.tonight...east winds less than 10 knots backing north after
midnight, then veering northeast late. a slight chance of showers
after midnight. waves calm to 2 feet. 
.thursday...northeast winds less than 10 knots increasing to 5 to
15 knots in the afternoon. waves calm to 2 feet. 
.thursday night...northeast winds less than 10 knots increasing
to 5 to 15 knots after midnight. waves calm to 2 feet. 
.friday...northeast winds 5 to 15 knots increasing to 15 to
25 knots by mid afternoon. waves calm to 2 feet. 
.friday night...east winds 10 to 20 knots diminishing to 5 to
15 knots after midnight. waves calm to 2 feet. 
.saturday...east winds 5 to 15 knots backing northwest. a chance
of showers. waves calm to 2 feet. 
.sunday...west winds 5 to 15 knots. waves calm to 2 feet. 

$$

fzus63 kmqt 120801
glfls 
expires:202608122000;;045626

open lake forecast for lake superior
national weather service marquette mi
401 am edt wed aug 12 2026

for waters beyond 5 nautical miles of shore on lake superior.

waves are the significant wave height - the average of the
highest 1/3 of the wave spectrum. occasional wave height is the
average of the highest 1/10 of the wave spectrum.

.synopsis...

this morning, 29.5 inch low pressure is over the gulf of st lawrence 
and deepens to 29.5 inches while a plains low of 29.7 inches is only 
separated from the first low by weak 29.8-29.9 inch ridging over 
lake superior. this pattern holds until thursday morning, when a 
30.2 inch high over the canadian rockies begins to extend stronger 
ridging over lake superior before eventually becoming centered over 
the lake at 30.2 inches friday afternoon. a 29.9 inch trough then 
moves over the western end of the lake saturday afternoon as the 
30.2 inch high moves over the lower great lakes. this trough 
consolidates to a 29.8 inch low as it passes over lake superior, 
reaching ontario by sunday afternoon, with 30.1 inch ridging to 
follow.




lsz264-122000-
lake superior from upper entrance to portage canal to manitou
island mi 5nm off shore to the us/canadian border-
401 am edt wed aug 12 2026

.early this morning...west winds less than 10 knots. waves calm
to 2 feet. 
.today...west winds less than 10 knots veering northwest in the
afternoon, then backing southwest late. waves calm to 2 feet. 
.tonight...southwest winds less than 10 knots veering northwest
late. waves calm to 2 feet. 
.thursday...north winds less than 10 knots veering northeast in
the afternoon, then veering southwest late. waves calm to 2 feet.
.thursday night...north winds less than 10 knots increasing to
5 to 15 knots after midnight. waves calm to 2 feet. 
.friday...northeast winds 5 to 15 knots veering east by mid
afternoon. waves calm to 2 feet. 
.friday night...southeast winds 5 to 15 knots. waves calm to
2 feet. 
.saturday...southeast winds 10 to 20 knots becoming northwest
5 to 15 knots. a chance of showers. waves calm to 2 feet. 
.sunday...northwest winds 5 to 15 knots backing west. waves calm
to 2 feet. 

$$

fzus63 kmqt 120801
glfls 
expires:202608122000;;045627

open lake forecast for lake superior
national weather service marquette mi
401 am edt wed aug 12 2026

for waters beyond 5 nautical miles of shore on lake superior.

waves are the significant wave height - the average of the
highest 1/3 of the wave spectrum. occasional wave height is the
average of the highest 1/10 of the wave spectrum.

.synopsis...

this morning, 29.5 inch low pressure is over the gulf of st lawrence 
and deepens to 29.5 inches while a plains low of 29.7 inches is only 
separated from the first low by weak 29.8-29.9 inch ridging over 
lake superior. this pattern holds until thursday morning, when a 
30.2 inch high over the canadian rockies begins to extend stronger 
ridging over lake superior before eventually becoming centered over 
the lake at 30.2 inches friday afternoon. a 29.9 inch trough then 
moves over the western end of the lake saturday afternoon as the 
30.2 inch high moves over the lower great lakes. this trough 
consolidates to a 29.8 inch low as it passes over lake superior, 
reaching ontario by sunday afternoon, with 30.1 inch ridging to 
follow.




lsz265-122000-
lake superior west of line from manitou island to marquette mi
beyond 5nm from shore-
401 am edt wed aug 12 2026

.early this morning...west winds 5 to 15 knots. waves calm to
2 feet. 
.today...west winds 5 to 15 knots veering northeast in the
afternoon, then veering east late. waves calm to 2 feet. 
.tonight...southeast winds 5 to 15 knots backing north late. a
slight chance of showers after midnight. waves calm to 2 feet. 
.thursday...north winds 5 to 15 knots veering east late. a slight
chance of showers through early afternoon. waves calm to 2 feet. 
.thursday night...north winds 5 to 15 knots. waves calm to
2 feet. 
.friday...northeast winds 5 to 15 knots. waves calm to 2 feet. 
.friday night...southeast winds 5 to 15 knots. waves calm to
2 feet. 
.saturday...southeast winds 15 to 25 knots diminishing to 10 to
20 knots. a chance of showers. waves building to 1 to 3 feet,
then subsiding to calm to 2 feet. 
.sunday...northwest winds 15 to 25 knots becoming west 10 to
20 knots. waves building to 1 to 3 feet, then subsiding to calm
to 2 feet. 

$$

fzus63 kmqt 120801
glfls 
expires:202608122000;;045628

open lake forecast for lake superior
national weather service marquette mi
401 am edt wed aug 12 2026

for waters beyond 5 nautical miles of shore on lake superior.

waves are the significant wave height - the average of the
highest 1/3 of the wave spectrum. occasional wave height is the
average of the highest 1/10 of the wave spectrum.

.synopsis...

this morning, 29.5 inch low pressure is over the gulf of st lawrence 
and deepens to 29.5 inches while a plains low of 29.7 inches is only 
separated from the first low by weak 29.8-29.9 inch ridging over 
lake superior. this pattern holds until thursday morning, when a 
30.2 inch high over the canadian rockies begins to extend stronger 
ridging over lake superior before eventually becoming centered over 
the lake at 30.2 inches friday afternoon. a 29.9 inch trough then 
moves over the western end of the lake saturday afternoon as the 
30.2 inch high moves over the lower great lakes. this trough 
consolidates to a 29.8 inch low as it passes over lake superior, 
reaching ontario by sunday afternoon, with 30.1 inch ridging to 
follow.




lsz266-122000-
lake superior east of a line from manitou island to marquette mi
and west of a line from grand marais mi to the us/canadian border
beyond 5nm from shore-
401 am edt wed aug 12 2026

.early this morning...west winds 5 to 15 knots. waves calm to
2 feet. 
.today...west winds 5 to 15 knots veering north late. waves calm
to 2 feet. 
.tonight...west winds 5 to 15 knots veering northwest late. waves
calm to 2 feet. 
.thursday...northwest winds 5 to 15 knots veering north in the
afternoon. waves calm to 2 feet. 
.thursday night...north winds 5 to 15 knots. waves calm to
2 feet. 
.friday...northeast winds 5 to 15 knots backing north by mid
afternoon. waves calm to 2 feet. 
.friday night...east winds 5 to 15 knots veering southeast after
midnight. waves calm to 2 feet. 
.saturday...southeast winds 15 to 25 knots becoming south 10 to
20 knots. a chance of showers. waves building to 1 to 3 feet,
then subsiding to calm to 2 feet. 
.sunday...northwest winds 15 to 25 knots diminishing to 10 to
20 knots. waves building to 2 to 4 feet, then subsiding to calm
to 2 feet. 

$$

fzus63 kmqt 120801
glfls 
expires:202608122000;;045629

open lake forecast for lake superior
national weather service marquette mi
401 am edt wed aug 12 2026

for waters beyond 5 nautical miles of shore on lake superior.

waves are the significant wave height - the average of the
highest 1/3 of the wave spectrum. occasional wave height is the
average of the highest 1/10 of the wave spectrum.

.synopsis...

this morning, 29.5 inch low pressure is over the gulf of st lawrence 
and deepens to 29.5 inches while a plains low of 29.7 inches is only 
separated from the first low by weak 29.8-29.9 inch ridging over 
lake superior. this pattern holds until thursday morning, when a 
30.2 inch high over the canadian rockies begins to extend stronger 
ridging over lake superior before eventually becoming centered over 
the lake at 30.2 inches friday afternoon. a 29.9 inch trough then 
moves over the western end of the lake saturday afternoon as the 
30.2 inch high moves over the lower great lakes. this trough 
consolidates to a 29.8 inch low as it passes over lake superior, 
reaching ontario by sunday afternoon, with 30.1 inch ridging to 
follow.




lsz267-122000-
lake superior from grand marais mi to whitefish point mi 5nm off
shore to the us/canadian border-
401 am edt wed aug 12 2026

.early this morning...northwest winds less than 10 knots. waves
calm to 2 feet. 
.today...northwest winds 5 to 15 knots. waves calm to 2 feet. 
.tonight...northwest winds 5 to 15 knots backing west late. waves
calm to 2 feet. 
.thursday...northwest winds 5 to 15 knots. waves calm to 2 feet. 
.thursday night...northwest winds 10 to 20 knots becoming north
5 to 15 knots after midnight. waves calm to 2 feet. 
.friday...north winds 5 to 15 knots. waves calm to 2 feet. 
.friday night...north winds 5 to 15 knots veering southeast after
midnight. waves calm to 2 feet. 
.saturday...southeast winds 10 to 20 knots. showers likely. waves
calm to 2 feet. 
.sunday...northwest winds 10 to 20 knots diminishing to 5 to
15 knots. waves building to 1 to 3 feet, then subsiding to calm
to 2 feet. 

$$