Battle of Yarmuk
George F. Nafziger, in his book Islam at war, described the battle:

Although Yarmouk is little known today, it is one of the most decisive battles in human history...... Had Heraclius' forces prevailed, the modern world would be so changed as to be unrecognizable.

Notes: ^ a: Modern estimates for Roman army:
Donner 1981: 20,000–40,000.
Nicolle 1994: (p. 32) Maximum 25,000 with an unstated number of Arab reinforcements.  (p. 65) 15,000–20,000.
Akram 1970: 150,000.
Kaegi 2003 (p. 242): 15,000–20,000, possibly more
Kennedy 2006 (p. 131): 15,000–20,000, 
^ b: Roman source for Roman army:
Theophanes (pp. 337–338): 80,000 Roman troops (Kennedy, 2006, p. 145) and 60,000 allied Ghassanid troops (Gibbon, Vol. 5, p. 325).
^ c: Early Muslim sources for Roman army:
Baladhuri (p. 140): 200,000.
Tabari (Vol. 2, p. 598): 200,000.
Ibn Ishaq (Tabari, Vol. 3, p. 75): 100,000 against 24,000 Muslims.
^ d: Modern estimates for Muslim army:
Kaegi 1995: 15,000–20,000 maximum
Nicolle 1994: (p. 43) 20,000–40,000, likely 25,000.
Akram: 40,000 maximum.
Treadgold 1997: 24,000

^ e: Primary sources for Muslim army:
Ibn Ishaq (Vol. 3, p. 74): 24,000.
Baladhuri: 24,000.
Tabari (Vol. 2, p. 592): 40,000.
^ f: Primary sources for Roman casualties:
Tabari (Vol. 2, p. 596): 120,000 killed.
Ibn Ishaq (Vol. 3, p. 75): 70,000 killed.
Baladhuri (p. 141): 70,000 killed.
^ g: His name is mentioned in Islamic sources as Jaban, Vahan Benaas and Mahan. Vahan is most likely to be his name as it is of Armenian origin
^ i: During the reign of Abu Bakr, Khalid ibn Walid remained the Commander-in-Chief of the army in Syria but at Umar's accession as Caliph he dismissed him from command. Abu Ubaidah ibn al-Jarrah became the new commander in chief. (See Dismissal of Khalid).
^ j: Some Byzantine sources also mention a fortified encampment at Yaqusah, 18 kilometres (11 mi) from the battlefield. E.g., A. I. Akram suggests that the Byzantine camps were north of Wadi-ur-Ruqqad, while David Nicolle agrees with early Armenian sources, which positioned camps at Yaqusah (See: Nicolle p. 61 and Akram 2004 p. 410).
^ k: Akram misinterprets the bridge at 'Ayn Dhakar for a ford while Nicolle explains the exact geography (See: Nicolle p. 64 and Akram p. 410)
^ n: Concepts used in the description of the battle lines of the Muslims and the Byzantines. See image-1.

References: 

Bibliography: 

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Futuh al Sham (The Islamic Conquest of Syria). Translated by al-Kindi, Mawlana Sulayman. Ta-Ha Publishers. 2002.
Chronicle of Fredegar. 658.
Dionysius Telmaharensis (774). Chronicle of Pseudo-Dionysius of Tell-Mahre.
Ibn Ishaq (2004). Sirah Rasul Allah [The Life of Muhammad]. Translated by Guillaume, Alfred. Oxford and New York: Oxford University Press. ISBN 9780196360331.
Ibn Khaldun (2020). Nessim Joseph, Dawood (ed.). Muqaddimah [Introduction]. Translated by Rosenthal, Franz. Princeton and Oxford: Princeton University Press. doi:10.2307/j.ctvwh8dcw. ISBN 9781400866090. JSTOR j.ctvwh8dcw.
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Muhammad ibn Jarir al-Tabari (1987). Yarshater, Ehsan (ed.). History of the Prophets and Kings. SUNY series in Near Eastern Studies. Vol. 2: Prophets and Patriarchs. Translated by Brinner, William M. Albany, NY: State University of New York Press. doi:10.2307/jj.18252170. ISBN 9780887063138. JSTOR jj.18252170.
Muhammad ibn Jarir al-Tabari (2015). Yarshater, Ehsan (ed.). History of the Prophets and Kings. SUNY series in Near Eastern Studies. Vol. 2: The Children of Israel. Translated by Brinner, William M. Albany, NY: State University of New York Press. doi:10.2307/jj.18252188. ISBN 9780791497524. JSTOR jj.18252188.
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Journals: Conrad, Lawrence I. (1988). "Seven and the Tasbīʿ: On the Implications of Numerical Symbolism for the Study of Medieval Islamic History". Journal of the Economic and Social History of the Orient. 31 (1): 42–73. doi:10.2307/3631765. JSTOR 3631765.
Jandora, John W. (1986). "Developments in Islamic Warfare: The Early Conquests". Studia Islamica (64): 101–113. doi:10.2307/1596048. JSTOR 1596048.

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Donner, Fred McGraw (1981). The Early Islamic Conquests. Princeton Studies on the Near East. Vol. 1017. Princeton University Press. doi:10.1515/9781400847877. ISBN 9780691053271.
Greatrex, Geoffrey; Lieu, Samuel N.C. (2002). The Roman Eastern Frontier and the Persian Wars (Part II, 363–628 AD). Oxford and New York: Routledge. doi:10.4324/9780203994542. ISBN 9780203994542.
Gil, Moshe (1997). A History of Palestine, 634–1099. Translated by Broido, Ethel. Cambridge and New York: Cambridge University Press. ISBN 9780521599849.
Haldon, John (1997). Byzantium in the Seventh Century: the Transformation of a Culture. Cambridge and New York: Cambridge University Press. doi:10.1017/CBO9780511582318. ISBN 9780521319171.
Haldon, John (2008). The Byzantine Wars. Stroud: The History Press. ISBN 978-0-7524-9652-8.
Hoyland, Robert G. (1997). Seeing Islam as Others Saw It. Studies in late antiquity and early Islam.